A new modified global k-means algorithm for clustering large data sets

Wed 28 Apr 2021 12:28:10 AEST

The k-means algorithm and its variations are known to be fast clustering algorithms. However, they are sensitive to the choice of starting points and inefficient for solving clustering problems in large data sets. Recently, in order to resolve difficulties with the choice of starting points, incremental approaches have been developed. The modified global k-means algorithm is based on such an approach. It iteratively adds one cluster center at a time. Numerical experiments show that this algorithm considerably improve the k-means algorithm. However, this algorithm is not suitable for clustering very large data sets. In this paper, a new version of the modified global k-means algorithm is proposed. We introduce an auxiliary cluster function to generate a set of starting points spanning different parts of the data set. We exploit information gathered in previous iterations of the incremental algorithm to reduce its complexity.

An incremental approach for the construction of a piecewise linear classifier

Wed 28 Apr 2021 12:24:03 AEST

In this paper the problem of finding piecewise linear boundaries between sets is considered and is applied for solving supervised data classification problems. An algorithm for the computation of piecewise linear boundaries, consisting of two main steps, is proposed. In the first step sets are approximated by hyperboxes to find so-called “indeterminate” regions between sets. In the second step sets are separated inside these “indeterminate” regions by piecewise linear functions. These functions are computed incrementally starting with a linear function. Results of numerical experiments are reported. These results demonstrate that the new algorithm requires a reasonable training time and it produces consistently good test set accuracy on most data sets comparing with mainstream classifiers.

Discrete gradient methods

Wed 07 Apr 2021 14:02:47 AEST

In this chapter, the notion of a discrete gradient is introduced and it is shown that the discrete gradients can be used to approximate subdifferentials of a broad class of nonsmooth functions. Two methods based on such approximations, more specifically, the discrete gradient method (DGM) and its limited memory version (LDGB), are described. These methods are semi derivative-free methods for solving nonsmooth and, in general, nonconvex optimization problems. The performance of the methods is demonstrated using some academic test problems. © Springer Nature Switzerland AG 2020.

Final words

Wed 07 Apr 2021 14:02:47 AEST

Introduction

Wed 07 Apr 2021 14:02:47 AEST

Nonsmooth optimization (NSO) is among the most challenging tasks in the field of mathematical programming. It addresses optimization problems where objective and/or constraint functions have discontinuous gradients. NSO problems arise in many real life applications. Moreover, some smooth optimization techniques like different decomposition methods, dual formulations and exact penalty methods may lead us to solve NSO problems being either smaller in dimension or simpler in structure. In addition, some optimization problems may be analytically smooth but numerically nonsmooth. This is the case, for instance, with noisy input data and so-called stiff problems, which are numerically unstable and behave like nonsmooth problems. © Springer Nature Switzerland AG 2020.

Prediction of gold-bearing localised occurrences from limited exploration data

Wed 07 Apr 2021 14:02:34 AEST

Inaccurate drill-core assay interpretation in the exploration stage presents challenges to long-term profit of gold mining operations. Predicting the gold distribution within a deposit as precisely as possible is one of the most important aspects of the methodologies employed to avoid problems associated with financial expectations. The prediction of the variability of gold using a very limited number of drill-core samples is a very challenging problem. This is often intractable using traditional statistical tools where with less than complete spatial information certain assumptions are made about gold distribution and mineralisation. The decision-support predictive modelling methodology based on the unsupervised machine learning technique, presented in this paper avoids some of the restrictive limitations of traditional methods. It identifies promising exploration targets missed during exploration and recovers hidden spatial and physical characteristics of the explored deposit using information directly from drill hole database. Copyright © 2020 Inderscience Enterprises Ltd.

An approximate ADMM for solving linearly constrained nonsmooth optimization problems with two blocks of variables

Wed 07 Apr 2021 14:01:58 AEST

Nonsmooth convex optimization problems with two blocks of variables subject to linear constraints are considered. A new version of the alternating direction method of multipliers is developed for solving these problems. In this method the subproblems are solved approximately. The convergence of the method is studied. New test problems are designed and used to verify the efficiency of the proposed method and to compare it with two versions of the proximal bundle method.

A difference of convex optimization algorithm for piecewise linear regression

Wed 07 Apr 2021 14:01:56 AEST

The problem of finding a continuous piecewise linear function approximating a regression function is considered. This problem is formulated as a nonconvex nonsmooth optimization problem where the objective function is represented as a difference of convex (DC) functions. Subdifferentials of DC components are computed and an algorithm is designed based on these subdifferentials to find piecewise linear functions. The algorithm is tested using some synthetic and real world data sets and compared with other regression algorithms.

A sharp augmented Lagrangian-based method in constrained non-convex optimization

Wed 07 Apr 2021 14:01:56 AEST

In this paper, a novel sharp Augmented Lagrangian-based global optimization method is developed for solving constrained non-convex optimization problems. The algorithm consists of outer and inner loops. At each inner iteration, the discrete gradient method is applied to minimize the sharp augmented Lagrangian function. Depending on the solution found the algorithm stops or updates the dual variables in the inner loop, or updates the upper or lower bounds by going to the outer loop. The convergence results for the proposed method are presented. The performance of the method is demonstrated using a wide range of nonlinear smooth and non-smooth constrained optimization test problems from the literature.

A server side solution for detecting webInject : A machine learning approach

Wed 07 Apr 2021 14:01:54 AEST

With the advancement of client-side on the fly web content generation techniques, it becomes easier for attackers to modify the content of a website dynamically and gain access to valuable information. A majority portion of online attacks is now done by WebInject. The end users are not always skilled enough to differentiate between injected content and actual contents of a webpage. Some of the existing solutions are designed for client side and all the users have to install it in their system, which is a challenging task. In addition, various platforms and tools are used by individuals, so different solutions needed to be designed. Existing server side solution often focuses on sanitizing and filtering the inputs. It will fail to detect obfuscated and hidden scripts. In this paper, we propose a server side solution using a machine learning approach to detect WebInject in banking websites. Unlike other techniques, our method collects features of a Document Object Model (DOM) and classifies it with the help of a pre-trained model.

Clustering in large data sets with the limited memory bundle method

Wed 07 Apr 2021 14:01:27 AEST

The aim of this paper is to design an algorithm based on nonsmooth optimization techniques to solve the minimum sum-of-squares clustering problems in very large data sets. First, the clustering problem is formulated as a nonsmooth optimization problem. Then the limited memory bundle method [Haarala et al., 2007] is modified and combined with an incremental approach to design a new clustering algorithm. The algorithm is evaluated using real world data sets with both the large number of attributes and the large number of data points. It is also compared with some other optimization based clustering algorithms. The numerical results demonstrate the efficiency of the proposed algorithm for clustering in very large data sets.

Solving minimax problems : Local smoothing versus global smoothing

Wed 07 Apr 2021 14:01:26 AEST

The aim of this chapter is to compare different smoothing techniques for solving finite minimax problems. We consider the local smoothing technique which approximates the function in some neighborhood of a point of nondifferentiability and also global smoothing techniques such as the exponential and hyperbolic smoothing which approximate the function in the whole domain. Computational results on the collection of academic test problems are used to compare different smoothing techniques. Results show the superiority of the local smoothing technique for convex problems and global smoothing techniques for nonconvex problems. © 2018, Springer International Publishing AG, part of Springer Nature.

Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations

Wed 07 Apr 2021 14:01:15 AEST

We introduce a proximal bundle method for the numerical minimization of a nonsmooth difference-of-convex (DC) function. Exploiting some classic ideas coming from cutting-plane approaches for the convex case, we iteratively build two separate piecewise-affine approximations of the component functions, grouping the corresponding information in two separate bundles. In the bundle of the first component, only information related to points close to the current iterate are maintained, while the second bundle only refers to a global model of the corresponding component function. We combine the two convex piecewise-affine approximations, and generate a DC piecewise-affine model, which can also be seen as the pointwise maximum of several concave piecewise-affine functions. Such a nonconvex model is locally approximated by means of an auxiliary quadratic program, whose solution is used to certify approximate criticality or to generate a descent search-direction, along with a predicted reduction, that is next explored in a line-search setting. To improve the approximation properties at points that are far from the current iterate a supplementary quadratic program is also introduced to generate an alternative more promising search-direction. We discuss the main convergence issues of the line-search based proximal bundle method, and provide computational results on a set of academic benchmark test problems. © 2017, Springer Science+Business Media, LLC.

A comparative assessment of models to predict monthly rainfall in Australia

Wed 07 Apr 2021 14:01:14 AEST

Accurate rainfall prediction is a challenging task. It is especially challenging in Australia where the climate is highly variable. Australia’s climatic zones range from high rainfall tropical regions in the north to the driest desert region in the interior. The performance of prediction models may vary depending on climatic conditions. It is, therefore, important to assess and compare the performance of these models in different climatic zones. This paper examines the performance of data driven models such as the support vector machines for regression, the multiple linear regression, the k-nearest neighbors and the artificial neural networks for monthly rainfall prediction in Australia depending on climatic conditions. Rainfall data with five meteorological variables over the period of 1970–2014 from 24 geographically diverse weather stations are used for this purpose. The prediction performance of each model was evaluated by comparing observed and predicted rainfall using various measures for prediction accuracy. © 2018, Springer Science+Business Media B.V., part of Springer Nature.

Nonsmooth DC programming approach to clusterwise linear regression : Optimality conditions and algorithms

Wed 07 Apr 2021 14:01:11 AEST

The clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem using the squared regression error function. The objective function in this problem is represented as a difference of convex functions. Optimality conditions are derived, and an algorithm is designed based on such a representation. An incremental approach is proposed to generate starting solutions. The algorithm is tested on small to large data sets. © 2017 Informa UK Limited, trading as Taylor & Francis Group.

Batch clustering algorithm for big data sets

Wed 07 Apr 2021 14:01:11 AEST

Vast spread of computing technologies has led to abundance of large data sets. Today tech companies like, Google, Facebook, Twitter and Amazon handle big data sets and log terabytes, if not petabytes, of data per day. Thus, there is a need to find similarities and define groupings among the elements of these big data sets. One of the ways to find these similarities is data clustering. Currently, there exist several data clustering algorithms which differ by their application area and efficiency. Increase in computational power and algorithmic improvements have reduced the time for clustering of big data sets. But it usually happens that big data sets can't be processed whole due to hardware and computational restrictions. In this paper, the classic k-means clustering algorithm is compared to the proposed batch clustering (BC) algorithm for the required computation time and objective function. The BC algorithm is designed to cluster large data sets in batches but maintain the efficiency and quality. Several experiments confirm that batch clustering algorithm for big data sets is more efficient in using computational power, data storage and results in better clustering compared to k-means algorithm. The experiments are conducted with the data set of 2 (two) million two-dimensional data points. © 2016 IEEE.

A new modification of Kohonen neural network for VQ and clustering problems

Wed 07 Apr 2021 14:00:38 AEST

Vector Quantization (VQ) and Clustering are significantly important in the field of data mining and pattern recognition. The Self Organizing Maps has been widely used for clustering and topology visualization. The topology of the SOM and its initial neurons play an important role in the convergence of the Kohonen neural network. In this paper, in order to improve the convergence of the SOM we introduce an algorithm based on the split and merging of clusters to initialize neurons. We also introduce a topology based on this initialization to optimize the vector quantization error. Such an approach allows one to find global or near global solution to the vector quantization and consequently clustering problem. The numerical results on 4 small to large real-world data sets are reported to demonstrate the performance of the proposed algorithm.

An algorithm for clustering using L1-norm based on hyperbolic smoothing technique

Wed 07 Apr 2021 14:00:32 AEST

Cluster analysis deals with the problem of organization of a collection of objects into clusters based on a similarity measure, which can be defined using various distance functions. The use of different similarity measures allows one to find different cluster structures in a data set. In this article, an algorithm is developed to solve clustering problems where the similarity measure is defined using the L1-norm. The algorithm is designed using the nonsmooth optimization approach to the clustering problem. Smoothing techniques are applied to smooth both the clustering function and the L1-norm. The algorithm computes clusters sequentially and finds global or near global solutions to the clustering problem. Results of numerical experiments using 12 real-world data sets are reported, and the proposed algorithm is compared with two other clustering algorithms. ©2015 Wiley Periodicals, Inc.

Optimization based clustering algorithms for authorship analysis of phishing emails

Wed 07 Apr 2021 14:00:29 AEST

Phishing has given attackers power to masquerade as legitimate users of organizations, such as banks, to scam money and private information from victims. Phishing is so widespread that combating the phishing attacks could overwhelm the victim organization. It is important to group the phishing attacks to formulate effective defence mechanism. In this paper, we use clustering methods to analyze and characterize phishing emails and perform their relative attribution. Emails are first tokenized to a bag-of-word space and, then, transformed to a numeric vector space using frequencies of words in documents. Wordnet vocabulary is used to take effects of similar words into account and to reduce sparsity. The word similarity measure is combined with the term frequencies to introduce a novel text transformation into numeric features. To improve the accuracy, we apply inverse document frequency weighting, which gives higher weights to features used by fewer authors. The k-means and recently introduced three optimization based algorithms: MS-MGKM, INCA and DCClust are applied for clustering purposes. The optimization based algorithms indicate the existence of well separated clusters in the phishing emails dataset. © 2017, Springer Science+Business Media New York.

A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes

Wed 07 Apr 2021 13:57:14 AEST

In this paper, we develop a version of the bundle method to solve unconstrained difference of convex (DC) programming problems. It is assumed that a DC representation of the objective function is available. Our main idea is to utilize subgradients of both the first and second components in the DC representation. This subgradient information is gathered from some neighborhood of the current iteration point and it is used to build separately an approximation for each component in the DC representation. By combining these approximations we obtain a new nonconvex cutting plane model of the original objective function, which takes into account explicitly both the convex and the concave behavior of the objective function. We design the proximal bundle method for DC programming based on this new approach and prove the convergence of the method to an -critical point. The algorithm is tested using some academic test problems and the preliminary numerical results have shown the good performance of the new bundle method. An interesting fact is that the new algorithm finds nearly always the global solution in our test problems.

New diagonal bundle method for clustering problems in large data sets

Wed 07 Apr 2021 13:57:02 AEST

Clustering is one of the most important tasks in data mining. Recent developments in computer hardware allow us to store in random access memory (RAM) and repeatedly read data sets with hundreds of thousands and even millions of data points. This makes it possible to use conventional clustering algorithms in such data sets. However, these algorithms may need prohibitively large computational time and fail to produce accurate solutions. Therefore, it is important to develop clustering algorithms which are accurate and can provide real time clustering in large data sets. This paper introduces one of them. Using nonsmooth optimization formulation of the clustering problem the objective function is represented as a difference of two convex (DC) functions. Then a new diagonal bundle algorithm that explicitly uses this structure is designed and combined with an incremental approach to solve this problem. The method is evaluated using real world data sets with both large number of attributes and large number of data points. The proposed method is compared with two other clustering algorithms using numerical results. © 2017 Elsevier B.V.

Aggregate codifferential method for nonsmooth DC optimization

Wed 07 Apr 2021 13:56:39 AEST

A new algorithm is developed based on the concept of codifferential for minimizing the difference of convex nonsmooth functions. Since the computation of the whole codifferential is not always possible, we use a fixed number of elements from the codifferential to compute the search directions. The convergence of the proposed algorithm is proved. The efficiency of the algorithm is demonstrated by comparing it with the subgradient, the truncated codifferential and the proximal bundle methods using nonsmooth optimization test problems.

CR-Modified SOM to the problem of handwritten digits recognition

Wed 07 Apr 2021 13:56:37 AEST

Recently, researchers show that the handwritten digit recognition is a challenging problem. In this paper first, we introduce a Modified Self Organizing Maps for vector quantization problem then we present a Convolutional Recursive ModifiedSOMto the problem of handwritten digit recognition. TheModifiedSOMis novel in the sense of initialization process and the topology preservation. The experimental result on the well known digit database of MNIST, denotes the superiority of the proposed algorithm over the existing SOM-based methods.

Prediction of monthly rainfall in Victoria, Australia : Clusterwise linear regression approach

Wed 07 Apr 2021 13:56:34 AEST

This paper develops the Clusterwise Linear Regression (CLR) technique for prediction of monthly rainfall. The CLR is a combination of clustering and regression techniques. It is formulated as an optimization problem and an incremental algorithm is designed to solve it. The algorithm is applied to predict monthly rainfall in Victoria, Australia using rainfall data with five input meteorological variables over the period of 1889–2014 from eight geographically diverse weather stations. The prediction performance of the CLR method is evaluated by comparing observed and predicted rainfall values using four measures of forecast accuracy. The proposed method is also compared with the CLR using the maximum likelihood framework by the expectation-maximization algorithm, multiple linear regression, artificial neural networks and the support vector machines for regression models using computational results. The results demonstrate that the proposed algorithm outperforms other methods in most locations. © 2017 Elsevier B.V.

Piecewise linear classifiers based on nonsmooth optimization approaches

Wed 07 Apr 2021 13:56:34 AEST

Nonsmooth optimization provides efficient algorithms for solving many machine learning problems. In particular, nonsmooth optimization approaches to supervised data classification problems lead to the design of very efficient algorithms for their solution. In this chapter, we demonstrate how nonsmooth optimization algorithms can be applied to design efficient piecewise linear classifiers for supervised data classification problems. Such classifiers are developed using a max–min and a polyhedral conic separabilities as well as an incremental approach. We report results of numerical experiments and compare the piecewise linear classifiers with a number of other mainstream classifiers.

Constrained self organizing maps for data clusters visualization

Wed 07 Apr 2021 13:56:00 AEST

High dimensional data visualization is one of the main tasks in the field of data mining and pattern recognition. The self organizing maps (SOM) is one of the topology visualizing tool that contains a set of neurons that gradually adapt to input data space by competitive learning and form clusters. The topology preservation of the SOM strongly depends on the learning process. Due to this limitation one cannot guarantee the convergence of the SOM in data sets with clusters of arbitrary shape. In this paper, we introduce Constrained SOM (CSOM), the new version of the SOM by modifying the learning algorithm. The idea is to introduce an adaptive constraint parameter to the learning process to improve the topology preservation and mapping quality of the basic SOM. The computational complexity of the CSOM is less than those with the SOM. The proposed algorithm is compared with similar topology preservation algorithms and the numerical results on eight small to large real-world data sets demonstrate the efficiency of the proposed algorithm. © 2015, Springer Science+Business Media New York.

Sensitivity of algorithm parameters and objective function scaling in multi-objective optimisation of water distribution systems

Wed 07 Apr 2021 13:56:00 AEST

This paper presents an extensive analysis of the sensitivity of multi-objective algorithm parameters and objective function scaling tested on a large number of parameter setting combinations for a water distribution system optimisation problem. The optimisation model comprises two operational objectives minimised concurrently, the pump energy costs and deviations of constituent concentrations as a water quality measure. This optimisation model is applied to a regional nondrinking water distribution system, and solved using the optimisation software GANetXL incorporating the NSGA-II linked with the network analysis software EPANet. The sensitivity analysis employs a set of performance metrics, which were designed to capture the overall quality of the computed Pareto fronts. The performance and sensitivity of NSGA-II parameters using those metrics is evaluated. The results demonstrate that NSGA-II is sensitive to different parameter settings, and unlike in the single-objective problems, a range of parameter setting combinations appears to be required to reach a Pareto front of optimal solutions. Additionally, inadequately scaled objective functions cause the NSGA-II bias towards the second objective. Lastly, the methodology for performance and sensitivity analysis may be used for calibration of algorithm parameters.

Nonsmooth DC programming approach to the minimum sum-of-squares clustering problems

Wed 07 Apr 2021 13:55:52 AEST

This paper introduces an algorithm for solving the minimum sum-of-squares clustering problems using their difference of convex representations. A non-smooth non-convex optimization formulation of the clustering problem is used to design the algorithm. Characterizations of critical points, stationary points in the sense of generalized gradients and inf-stationary points of the clustering problem are given. The proposed algorithm is tested and compared with other clustering algorithms using large real world data sets. © 2015 Elsevier Ltd. All rights reserved.

Nonsmooth optimization based algorithms in cluster analysis

Wed 07 Apr 2021 13:55:38 AEST

Cluster analysis is an important task in data mining. It deals with the problem of organization of a collection of objects into clusters based on a similarity measure. Various distance functions can be used to define the similarity measure. Cluster analysis problems with the similarity measure defined by the squared Euclidean distance, which is also known as the minimum sum-of-squares clustering, has been studied extensively over the last five decades. L1 and L1 norms have attracted less attention. In this chapter, we consider a nonsmooth nonconvex optimization formulation of the cluster analysis problems. This formulation allows one to easily apply similarity measures defined using different distance functions. Moreover, an efficient incremental algorithm can be designed based on this formulation to solve the clustering problems. We develop incremental algorithms for solving clustering problems where the similarity measure is defined using the L1; L2 and L1 norms. We also consider different algorithms for solving nonsmooth nonconvex optimization problems in cluster analysis. The proposed algorithms are tested using several real world data sets and compared with other similar algorithms.

Introduction to Nonsmooth Optimization : Theory, practice and software

Wed 07 Apr 2021 13:55:38 AEST

This book is the first easy-to-read text on nonsmooth optimization (NSO, not necessarily differentiable optimization). Soving these kinds of problems plays a critical role in many industrial applications and real-world modeling systems, for example in the context of image denoising, optimal control, neural network training, data mining, ecomonics, and computational chemistry and physics. The book covers both the theory and the numerical methods used in NSO, and provides an overview of different problems arising in the field. It is organized into three parts: 1. convex and nonconvex analysis and the theory of NSO; 2. test problems and practical applications; 3. a guide to NSO software. The book is ideal for anyone teaching or attending NSO courses. As an accessible introduction to the field, it is also well suited as an independent learning guide for practitioners already familiar with the basics of optimization.

Impact of water-quality conditions in source reservoirs on the optimal operation of a regional multiquality water-distribution system

Wed 07 Apr 2021 13:55:37 AEST

The impact of water quality conditions in source reservoirs on the optimal operation of a regional multiquality water-distribution system is analyzed. The optimization model concurrently minimizes three operational objectives being pump energy costs, turbidity, and salinity deviations at customer demand nodes from allowed values. The optimization problem is solved using the optimization tool GANetXL incorporating the NSGA-II, linked with the network analysis software EPANet. The example network adapted from the literature captures some of the unique features of the Wimmera Mallee Pipeline in Australia. Six scenarios representing different water quality conditions in source reservoirs are analyzed. It was discovered that two types of trade-offs, competing and noncompeting, exist between the objectives and that the type of trade-off is not unique between a particular pair of objectives for all scenarios. These and other findings may be of particular use to system operators in their long-term operational planning and decision making. (C) 2015 American Society of Civil Engineers.

An incremental piecewise linear classifier based on polyhedral conic separation

Wed 07 Apr 2021 13:55:35 AEST

In this paper, a piecewise linear classifier based on polyhedral conic separation is developed. This classifier builds nonlinear boundaries between classes using polyhedral conic functions. Since the number of polyhedral conic functions separating classes is not known a priori, an incremental approach is proposed to build separating functions. These functions are found by minimizing an error function which is nonsmooth and nonconvex. A special procedure is proposed to generate starting points to minimize the error function and this procedure is based on the incremental approach. The discrete gradient method, which is a derivative-free method for nonsmooth optimization, is applied to minimize the error function starting from those points. The proposed classifier is applied to solve classification problems on 12 publicly available data sets and compared with some mainstream and piecewise linear classifiers. © 2014, The Author(s).

A history of water distribution systems and their optimisation

Wed 07 Apr 2021 13:54:52 AEST

Water distribution systems have a very long and rich history dating back to the third millennium B.C. Advances in water supply and distribution were followed in parallel by discoveries and inventions in other related fields. Therefore, it is the aim of this paper to review both the history of water distribution systems and those related fields in order to present a coherent summary of the complex multi-stranded discipline of water engineering. Related fields reviewed in this paper include devices for raising water and water pumps, water quality and water treatment, hydraulics, network analysis, and optimisation of water distribution systems. The review is brief and concise and allows the reader to quickly gain an understanding of the history and advancements of water distribution systems and analysis. Furthermore, the paper gives details of other existing publications where more information can be found.

Exploration of the trade-offs between water quality and pumping costs in optimal operation of regional multiquality water distribution systems

Wed 07 Apr 2021 13:54:52 AEST

This paper explores the trade-offs between water quality and pumping costs objectives in optimization of operation of regional multiquality water distribution systems. The optimization model is designed to concurrently minimize each objective, where water quality is represented by the deviations of constituent concentrations from required values and pumping costs are represented by energy consumed by the pumps. The optimization problem is solved using an optimization software, incorporating the nondominated sorting genetic algorithm II (NSGA-II), linked with network analysis software. Two typical but purposefully different example networks are used. First, a network with multiple water sources of different qualities and second, a network with one water source only, which was converted to represent a regional nondrinking water distribution system. The trade-offs between water quality and pumping costs are explored using a total of 14 scenarios reflecting different water quality configurations of these networks. Those scenarios, into which time variability was introduced for both source water quality and customer water quality requirements, were systematically developed to represent real-life situations that could be found in practice. The results indicate that for the majority of the scenarios, there is a trade-off with a competing nature between water quality and pumping costs objectives. Additionally, it was discovered that multiobjective optimization problems with water quality (i.e., concentration deviations) and pumping costs objectives could be reduced in certain instances into a single-objective problem of minimizing pumping costs. In fact, a regional water distribution system in which water quality is represented by a single conservative constituent can produce either a trade-off or single-objective solution between those two objectives, and this outcome is dependent on both the water quality configuration of the system and system operational flexibility. Last, some particular conclusions are drawn for both a water distribution system with multiple water sources and a water distribution system with a single water source, which suggest how changes in source water qualities or customer water quality requirements may impact system operation. It is, therefore, demonstrated that water utilities which operate regional multiquality nondrinking water distribution systems could benefit from the exploration of trade-offs between water quality and pumping costs for the purpose of operational planning.

Solving DC programs using the cutting angle method

Wed 07 Apr 2021 13:54:51 AEST

In this paper, we propose a new algorithm for global minimization of functions represented as a difference of two convex functions. The proposed method is a derivative free method and it is designed by adapting the extended cutting angle method. We present preliminary results of numerical experiments using test problems with difference of convex objective functions and box-constraints. We also compare the proposed algorithm with a classical one that uses prismatical subdivisions.

Nonsmooth optimization algorithm for solving clusterwise linear regression problems

Wed 07 Apr 2021 13:54:51 AEST

Clusterwise linear regression consists of finding a number of linear regression functions each approximating a subset of the data. In this paper, the clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem and an algorithm based on an incremental approach and on the discrete gradient method of nonsmooth optimization is designed to solve it. This algorithm incrementally divides the whole dataset into groups which can be easily approximated by one linear regression function. A special procedure is introduced to generate good starting points for solving global optimization problems at each iteration of the incremental algorithm. The algorithm is compared with the multi-start Spath and the incremental algorithms on several publicly available datasets for regression analysis.

An incremental clustering algorithm based on hyperbolic smoothing

Wed 07 Apr 2021 13:54:51 AEST

Clustering is an important problem in data mining. It can be formulated as a nonsmooth, nonconvex optimization problem. For the most global optimization techniques this problem is challenging even in medium size data sets. In this paper, we propose an approach that allows one to apply local methods of smooth optimization to solve the clustering problems. We apply an incremental approach to generate starting points for cluster centers which enables us to deal with nonconvexity of the problem. The hyperbolic smoothing technique is applied to handle nonsmoothness of the clustering problems and to make it possible application of smooth optimization algorithms to solve them. Results of numerical experiments with eleven real-world data sets and the comparison with state-of-the-art incremental clustering algorithms demonstrate that the smooth optimization algorithms in combination with the incremental approach are powerful alternative to existing clustering algorithms.

Modified self-organising maps with a new topology and initialisation algorithm

Wed 07 Apr 2021 13:54:30 AEST

Mapping quality of the self-organising maps (SOMs) is sensitive to the map topology and initialisation of neurons. In this article, in order to improve the convergence of the SOM, an algorithm based on split and merge of clusters to initialise neurons is introduced. The initialisation algorithm speeds up the learning process in large high-dimensional data sets. We also develop a topology based on this initialisation to optimise the vector quantisation error and topology preservation of the SOMs. Such an approach allows to find more accurate data visualisation and consequently clustering problem. The numerical results on eight small-to-large real-world data sets are reported to demonstrate the performance of the proposed algorithm in the sense of vector quantisation, topology preservation and CPU time requirement. © 2014 Taylor & Francis.

A heuristic algorithm for solving the minimum sum-of-squares clustering problems

Wed 07 Apr 2021 13:47:02 AEST

Clustering is an important task in data mining. It can be formulated as a global optimization problem which is challenging for existing global optimization techniques even in medium size data sets. Various heuristics were developed to solve the clustering problem. The global k-means and modified global k-means are among most efficient heuristics for solving the minimum sum-of-squares clustering problem. However, these algorithms are not always accurate in finding global or near global solutions to the clustering problem. In this paper, we introduce a new algorithm to improve the accuracy of the modified global k-means algorithm in finding global solutions. We use an auxiliary cluster problem to generate a set of initial points and apply the k-means algorithm starting from these points to find the global solution to the clustering problems. Numerical results on 16 real-world data sets clearly demonstrate the superiority of the proposed algorithm over the global and modified global k-means algorithms in finding global solutions to clustering problems.

An algorithm for clusterwise linear regression based on smoothing techniques

Wed 07 Apr 2021 13:46:58 AEST

We propose an algorithm based on an incremental approach and smoothing techniques to solve clusterwise linear regression (CLR) problems. This algorithm incrementally divides the whole data set into groups which can be easily approximated by one linear regression function. A special procedure is introduced to generate an initial solution for solving global optimization problems at each iteration of the incremental algorithm. Such an approach allows one to find global or approximate global solutions to the CLR problems. The algorithm is tested using several data sets for regression analysis and compared with the multistart and incremental Spath algorithms.

Truncated codifferential method for linearly constrained nonsmooth optimization

Wed 07 Apr 2021 13:46:41 AEST

In this paper a new algorithm is developed to minimize linearly constrained non-smooth optimization problem for convex objective functions. The algorithm is based on the concept of codifferential. The convergence of the proposed minimization algorithm is proved and results of numerical experiments using a set of test problems with nonsmooth convex objective function are reported.

Framework for multi-objective optimisation of the operation of water distribution networks including water quality

Wed 07 Apr 2021 13:46:34 AEST

Application of optimisation-based data mining techniques to medical data sets: A comparative analysis

Wed 07 Apr 2021 13:46:33 AEST

Abstract - Computational methods have become an important tool in the analysis of medical data sets. In this paper, we apply three optimisation-based data mining methods to the following data sets: (i) a cystic fibrosis data set and (ii) a tobacco control data set. Three algorithms used in the analysis of these data sets include: the modified linear least square fit, an optimization based heuristic algorithm for feature selection and an optimization based clustering algorithm. All these methods explore the relationship between features and classes, with the aim of determining contribution of specific features to the class outcome. However, the three algorithms are based on completely different approaches. We apply these methods to solve feature selection and classification problems. We also present comparative analysis of the algorithms using computational results. Results obtained confirm that these algorithms may be effectively applied to the analysis of other (bio)medical data sets

Pumping costs and water quality in the battlefield of optimal operation of water distribution networks

Wed 07 Apr 2021 13:46:27 AEST

Subgradient and bundle methods for nonsmooth optimization

Wed 07 Apr 2021 13:46:27 AEST

The nonsmooth optimization methods can mainly be divided into two groups: subgradient and bundle methods. Usually, when developing new algorithms and testing them, the comparison is made between similar kinds of methods. The goal of this work is to test and compare different bundle and subgradient methods as well as some hybrids of these two and/or some others. The test set included a large amount of different unconstrained nonsmooth minimization problems, e.g., convex and nonconvex problems, piecewise linear and quadratic problems, and problems with different sizes. Rather than foreground some method over the others, our aim is to get some insight on which method is suitable for certain types of problems.

Capped K-NN Editing in definition lacking environments

Wed 07 Apr 2021 13:46:17 AEST

While any input may be contributing, imprecise specification of class of data subdivided into classes identifies as rather common a source of noise. The misrepresentation may be characteristic of the data or be caused by forcing of a regression problem into the classification type. Consideration is given to examples of this nature, and an alternative is proposed. In the main part, the approach is based on a well-known technique of data treatment for noise using k-NN. The paper advances an editing technique designed around idea of variable number of authenticating instances. Test runs performed on publicly available and proprietary data demonstrate high retention ability of the new procedure without loss of classification accuracy. Noise reduction methods in a broader classification context are extensively surveyed.

A convolutional recursive modified Self Organizing Map for handwritten digits recognition

Wed 07 Apr 2021 13:46:04 AEST

It is well known that the handwritten digits recognition is a challenging problem. Different classification algorithms have been applied to solve it. Among them, the Self Organizing Maps (SOM) produced promising results. In this paper, first we introduce a Modified SOM for the vector quantization problem with improved initialization process and topology preservation. Then we develop a Convolutional Recursive Modified SOM and apply it to the problem of handwritten digits recognition. The computational results obtained using the well known MNIST dataset demonstrate the superiority of the proposed algorithm over the existing SOM-based algorithms.

Preface of the special issue OR: Connecting sciences supported by global optimization related to the 25th European conference on operational research (EURO XXV 2012)

Wed 07 Apr 2021 13:46:03 AEST

Machine learning algorithms for analysis of DNA data sets

Wed 07 Apr 2021 13:45:28 AEST

The applications of machine learning algorithms to the analysis of data sets of DNA sequences are very important. The present chapter is devoted to the experimental investigation of applications of several machine learning algorithms for the analysis of a JLA data set consisting of DNA sequences derived from non-coding segments in the junction of the large single copy region and inverted repeat A of the chloroplast genome in Eucalyptus collected by Australian biologists. Data sets of this sort represent a new situation, where sophisticated alignment scores have to be used as a measure of similarity. The alignment scores do not satisfy properties of the Minkowski metric, and new machine learning approaches have to be investigated. The authors' experiments show that machine learning algorithms based on local alignment scores achieve very good agreement with known biological classes for this data set. A new machine learning algorithm based on graph partitioning performed best for clustering of the JLA data set. Our novel k-committees algorithm produced most accurate results for classification. Two new examples of synthetic data sets demonstrate that the authors' k-committees algorithm can outperform both the Nearest Neighbour and k-medoids algorithms simultaneously.

A novel approach to optimal pump scheduling in water distribution systems

Wed 07 Apr 2021 13:44:46 AEST

The operation of a water distribution system is a complex task which involves scheduling of pumps, regulating water levels of storages, and providing satisfactory water quality to customers at required flow and pressure. Pump scheduling is one of the most important tasks of the operation of a water distribution system as it represents the major part of its operating costs. In this paper, a novel approach for modeling of pump scheduling to minimize energy consumption by pumps is introduced which uses pump's start/end run times as continuous variables. This is different from other approaches where binary integer variables for each hour are typically used which is considered very impractical from an operational perspective. The problem is formulated as a nonlinear programming problem and a new algorithm is developed for its solution. This algorithm is based on the combination of the grid search with the Hooke-Jeeves pattern search method. The performance of the algorithm is evaluated using literature test problems applying the hydraulic simulation model EPANet.

Comparison of metaheuristic algorithms for pump operation optimization

Wed 07 Apr 2021 13:44:46 AEST

Pumping cost constitutes the main part of the overall operating cost of water distribution systems. There are different optimization formulations of the pumping cost minimization problem including those with application of continuous and integer programming approaches. To date mainly various metaheuristics have been applied to solve this problem. However, the comprehensive comparison of those metaheuristics has not been done. Such a comparison is important to identify strengths and weaknesses of different algorithms which reflects on their performance. In this paper, we present a methodology for comparative analysis of widely used metaheuristics for solving the pumping cost minimization problem. This methodology includes the following comparison criteria: (a) the "optimal solution" obtained; (b) the efficiency; and (c) robustness. Algorithms applied are: particle swarm optimization, artificial bee colony and firefly algorithms. These algorithms were applied to one test problem available in the literature. The results obtained demonstrate that the artificial bee colony is the most robust and the firefly is the most efficient and accurate algorithm for this test problem. Funding :ARC

Hyperbolic smoothing function method for minimax problems

Wed 07 Apr 2021 13:44:41 AEST

In this article, an approach for solving finite minimax problems is proposed. This approach is based on the use of hyperbolic smoothing functions. In order to apply the hyperbolic smoothing we reformulate the objective function in the minimax problem and study the relationship between the original minimax and reformulated problems. We also study main properties of the hyperbolic smoothing function. Based on these results an algorithm for solving the finite minimax problem is proposed and this algorithm is implemented in general algebraic modelling system. We present preliminary results of numerical experiments with well-known nonsmooth optimization test problems. We also compare the proposed algorithm with the algorithm that uses the exponential smoothing function as well as with the algorithm based on nonlinear programming reformulation of the finite minimax problem. © 2013 Copyright Taylor and Francis Group, LLC.

Nonsmooth nonconvex optimization approach to clusterwise linear regression problems

Wed 07 Apr 2021 13:44:32 AEST

Clusterwise regression consists of finding a number of regression functions each approximating a subset of the data. In this paper, a new approach for solving the clusterwise linear regression problems is proposed based on a nonsmooth nonconvex formulation. We present an algorithm for minimizing this nonsmooth nonconvex function. This algorithm incrementally divides the whole data set into groups which can be easily approximated by one linear regression function. A special procedure is introduced to generate a good starting point for solving global optimization problems at each iteration of the incremental algorithm. Such an approach allows one to find global or near global solution to the problem when the data sets are sufficiently dense. The algorithm is compared with the multistart Späth algorithm on several publicly available data sets for regression analysis. © 2013 Elsevier B.V. All rights reserved.

A novel hybrid neural learning algorithm using simulated annealing and quasisecant method

Wed 07 Apr 2021 13:44:27 AEST

In this paper, we propose a hybrid learning algorithm for the single hidden layer feedforward neural networks (SLFNs) for data classification. The proposed hybrid algorithm is a two-phase learning algorithm and is based on the quasisecant and the simulated annealing methods. First, the weights between the hidden layer and the output layer nodes (output layer weights) are adjusted by the quasisecant algorithm. Then the simulated annealing is applied for global attribute weighting. The weights between the input layer and the hidden layer nodes are fixed in advance and are not included in the learning process. The proposed two-phase learning of the network is a novel idea and is different from that of the existing ones. The numerical results on some benchmark data sets are also reported and these results are promising. © 2011, Australian Computer Society, Inc.

An algorithm for minimization of pumping costs in water distribution systems using a novel approach to pump scheduling

Wed 07 Apr 2021 13:44:22 AEST

The operation of a water distribution system is a complex task which involves scheduling of pumps, regulating water levels of storages, and providing satisfactory water quality to customers at required flow and pressure. Pump scheduling is one of the most important tasks of the operation of a water distribution system as it represents the major part of its operating costs. In this paper, a novel approach for modeling of explicit pump scheduling to minimize energy consumption by pumps is introduced which uses the pump start/end run times as continuous variables, and binary integer variables to describe the pump status at the beginning of the scheduling period. This is different from other approaches where binary integer variables for each hour are typically used, which is considered very impractical from an operational perspective. The problem is formulated as a mixed integer nonlinear programming problem, and a new algorithm is developed for its solution. This algorithm is based on the combination of the grid search with the Hooke-Jeeves pattern search method. The performance of the algorithm is evaluated using literature test problems applying the hydraulic simulation model EPANet. © 2012 Elsevier Ltd.

Limited memory discrete gradient bundle method for nonsmooth derivative-free optimization

Wed 07 Apr 2021 13:44:21 AEST

Typically, practical nonsmooth optimization problems involve functions with hundreds of variables. Moreover, there are many practical problems where the computation of even one subgradient is either a difficult or an impossible task. In such cases derivative-free methods are the better (or only) choice since they do not use explicit computation of subgradients. However, these methods require a large number of function evaluations even for moderately large problems. In this article, we propose an efficient derivative-free limited memory discrete gradient bundle method for nonsmooth, possibly nonconvex optimization. The convergence of the proposed method is proved for locally Lipschitz continuous functions and the numerical experiments to be presented confirm the usability of the method especially for medium size and large-scale problems. © 2012 Copyright Taylor and Francis Group, LLC.

Subgradient Method for Nonconvex Nonsmooth Optimization

Wed 07 Apr 2021 13:44:16 AEST

In this paper, we introduce a new method for solving nonconvex nonsmooth optimization problems. It uses quasisecants, which are subgradients computed in some neighborhood of a point. The proposed method contains simple procedures for finding descent directions and for solving line search subproblems. The convergence of the method is studied and preliminary results of numerical experiments are presented. The comparison of the proposed method with the subgradient and the proximal bundle methods is demonstrated using results of numerical experiments. © 2012 Springer Science+Business Media, LLC.

Comparing different nonsmooth minimization methods and software

Wed 07 Apr 2021 13:44:01 AEST

Most nonsmooth optimization (NSO) methods can be divided into two main groups: subgradient methods and bundle methods. In this paper, we test and compare different methods from both groups as well as some methods which may be considered as hybrids of these two and/or some others. All the solvers tested are so-called general black box methods which, at least in theory, can be applied to solve almost all NSO problems. The test set includes a large number of unconstrained nonsmooth convex and nonconvex problems of different size. In particular, it includes piecewise linear and quadratic problems. The aim of this work is not to foreground some methods over the others but to get some insight on which method to select for certain types of problems. © 2012 Taylor and Francis Group, LLC.

Application of optimisation-based data mining techniques to tobacco control dataset

Wed 07 Apr 2021 13:43:49 AEST

Tobacco smoking is one of the leading causes of death around the world. Consequently, control of tobacco use is an important global public health issue. Tobacco control may be aided by development of theoretical and methodological frameworks for describing and understanding complex tobacco control systems. Linear regression and logistic regression are currently very popular statistical techniques for modeling and analyzing complex data in tobacco control systems. However, in tobacco markets, numerous interrelated factors nontrivially interact with tobacco control policies, such that policies and control outcomes are nonlinearly related.

Cluster analysis of a tobacco control data set

Wed 07 Apr 2021 13:43:48 AEST

Development of theoretical and methodological frameworks in data analysis is fundamental for modeling complex tobacco control systems. Following this idea, a new optimization based approach was introduced in the paper through two distinct methods: the modified linear least square fit and a heuristic algorithm for feature slection based on optimization-based methods have the potential to detect nonlinearity, and therefore to be more effective analysis tools of complex data set. In this study we evaluate the modified global k-means clustering algorithm by applying it to a massive set of real-time tobacco control survey data. Cluster analysis identified fixed and stable clusters in the studied data. These clusters correspond to groups of smokers with similar behaviour and the identification of these clusters may allow us to give recommendations on modification of existing tobacco control systems and on the design of future data acquistion surveys.

A novel piecewise linear classifier based on polyhedral conic and max-min separabilities

Wed 07 Apr 2021 13:43:43 AEST

In this paper, an algorithm for finding piecewise linear boundaries between pattern classes is developed. This algorithm consists of two main stages. In the first stage, a polyhedral conic set is used to identify data points which lie inside their classes, and in the second stage we exclude those points to compute a piecewise linear boundary using the remaining data points. Piecewise linear boundaries are computed incrementally starting with one hyperplane. Such an approach allows one to significantly reduce the computational effort in many large data sets. Results of numerical experiments are reported. These results demonstrate that the new algorithm consistently produces a good test set accuracy on most data sets comparing with a number of other mainstream classifiers. Â© 2011 Sociedad de EstadÃstica e InvestigaciÃ³n Operativa.

Classification through incremental max-min separability

Wed 07 Apr 2021 13:43:25 AEST

Piecewise linear functions can be used to approximate non-linear decision boundaries between pattern classes. Piecewise linear boundaries are known to provide efficient real-time classifiers. However, they require a long training time. Finding piecewise linear boundaries between sets is a difficult optimization problem. Most approaches use heuristics to avoid solving this problem, which may lead to suboptimal piecewise linear boundaries. In this paper, we propose an algorithm for globally training hyperplanes using an incremental approach. Such an approach allows one to find a near global minimizer of the classification error function and to compute as few hyperplanes as needed for separating sets. We apply this algorithm for solving supervised data classification problems and report the results of numerical experiments on real-world data sets. These results demonstrate that the new algorithm requires a reasonable training time and its test set accuracy is consistently good on most data sets compared with mainstream classifiers. © 2010 Springer-Verlag London Limited.

Codifferential method for minimizing nonsmooth DC functions

Wed 07 Apr 2021 13:43:24 AEST

In this paper, a new algorithm to locally minimize nonsmooth functions represented as a difference of two convex functions (DC functions) is proposed. The algorithm is based on the concept of codifferential. It is assumed that DC decomposition of the objective function is known a priori. We develop an algorithm to compute descent directions using a few elements from codifferential. The convergence of the minimization algorithm is studied and its comparison with different versions of the bundle methods using results of numerical experiments is given. © 2010 Springer Science+Business Media, LLC.

An efficient algorithm for the incremental construction of a piecewise linear classifier

Wed 07 Apr 2021 13:34:58 AEST

In this paper the problem of finding piecewise linear boundaries between sets is considered and is applied for solving supervised data classification problems. An algorithm for the computation of piecewise linear boundaries, consisting of two main steps, is proposed. In the first step sets are approximated by hyperboxes to find so-called "indeterminate" regions between sets. In the second step sets are separated inside these "indeterminate" regions by piecewise linear functions. These functions are computed incrementally starting with a linear function. Results of numerical experiments are reported. These results demonstrate that the new algorithm requires a reasonable training time and it produces consistently good test set accuracy on most data sets comparing with mainstream classifiers. Â© 2010 Elsevier B.V. All rights reserved.

Fast modified global k-means algorithm for incremental cluster construction

Wed 07 Apr 2021 13:34:52 AEST

The k-means algorithm and its variations are known to be fast clustering algorithms. However, they are sensitive to the choice of starting points and are inefficient for solving clustering problems in large datasets. Recently, incremental approaches have been developed to resolve difficulties with the choice of starting points. The global k-means and the modified global k-means algorithms are based on such an approach. They iteratively add one cluster center at a time. Numerical experiments show that these algorithms considerably improve the k-means algorithm. However, they require storing the whole affinity matrix or computing this matrix at each iteration. This makes both algorithms time consuming and memory demanding for clustering even moderately large datasets. In this paper, a new version of the modified global k-means algorithm is proposed. We introduce an auxiliary cluster function to generate a set of starting points lying in different parts of the dataset. We exploit information gathered in previous iterations of the incremental algorithm to eliminate the need of computing or storing the whole affinity matrix and thereby to reduce computational effort and memory usage. Results of numerical experiments on six standard datasets demonstrate that the new algorithm is more efficient than the global and the modified global k-means algorithms. Â© 2010 Elsevier Ltd. All rights reserved.

A generalized subgradient method with piecewise linear subproblem

Wed 07 Apr 2021 13:34:51 AEST

In this paper, a new version of the quasisecant method for nonsmooth nonconvex optimization is developed. Quasisecants are overestimates to the objective function in some neighborhood of a given point. Subgradients are used to obtain quasisecants. We describe classes of nonsmooth functions where quasisecants can be computed explicitly. We show that a descent direction with suffcient decrease must satisfy a set of linear inequalities. In the proposed algorithm this set of linear inequalities is solved by applying the subgradient algorithm to minimize a piecewise linear function. We compare results of numerical experiments between the proposed algorithm and subgradient method. Copyright Â© 2010 Watam Press.

Optimisation of operations of a water distribution system for reduced power usage

Wed 07 Apr 2021 13:34:43 AEST

There are many improvements to operation that can be made to a water distribution system once it has been constructed and placed in ground. Pipes and associated storages and pumps are typically designed to meet average peak daily demands, offer some capacity for growth, and also allow for some deterioration of performance over time. However, the 'as constructed' performance of the pipeline is invariably different to what was designed on paper, and this is particularly so for anything other than design flows, such as during times of water restrictions when there are significantly reduced flows. Because of this, there remain significant benefits to owners and operators for the adaptive and global optimisation of such systems. The present paper uses the Ouyen subsystem of the Northern Mallee Pipeline, in Victoria, as a case study for the development of an optimisation model. This has been done with the intent of using this model to reduce costs and provide better service to customers on this system. The Ouyen subsystem consists of 1600 km of trunk and distribution pipeline servicing an area of 456,000 Ha. The system includes 2 fixed speed pumps diverting water from the Murray River at Liparoo into two 150 ML balancing storages at Ouyen, 4 variable speed pumps feeding water from the balancing storages into the pipeline system, 2 variable speed pressure booster pumps and 5 town balancing storages. When considering all these components of the system, power consumption becomes an important part of the overall operation. The present paper considers a global optimisation model to minimise power consumption while maintaining reasonable performance of the system. The main components of the model are described including the network structure and the costs functions associated with the system. The final model presents the cost functions associated with the pump scheduling, including the penalties descriptions associated with maintaining appropriate storages levels and pressure bounds within the water distribution network.

Alexander Rubinov - An outstanding scholar

Wed 07 Apr 2021 13:34:13 AEST

A quasisecant method for minimizing nonsmooth functions

Wed 07 Apr 2021 13:33:57 AEST

We present an algorithm to locally minimize nonsmooth, nonconvex functions. In order to find descent directions, the notion of quasisecants, introduced in this paper, is applied. We prove that the algorithm converges to Clarke stationary points. Numerical results are presented demonstrating the applicability of the proposed algorithm to a wide variety of nonsmooth, nonconvex optimization problems. We also compare the proposed algorithm with the bundle method using numerical results.

Determining regularization parameters for derivative free neural learning

Wed 07 Apr 2021 13:33:45 AEST

Derivative free optimization methods have recently gained a lot of attractions for neural learning. The curse of dimensionality for the neural learning problem makes local optimization methods very attractive; however the error surface contains many local minima. Discrete gradient method is a special case of derivative free methods based on bundle methods and has the ability to jump over many local minima. There are two types of problems that are associated with this when local optimization methods are used for neural learning. The first type of problems is initial sensitivity dependence problem- that is commonly solved by using a hybrid model. Our early research has shown that discrete gradient method combining with other global methods such as evolutionary algorithm makes them even more attractive. These types of hybrid models have been studied by other researchers also. Another less mentioned problem is the problem of large weight values for the synaptic connections of the network. Large synaptic weight values often lead to the problem of paralysis and convergence problem especially when a hybrid model is used for fine tuning the learning task. In this paper we study and analyse the effect of different regularization parameters for our objective function to restrict the weight values without compromising the classification accuracy.

Comments on : Optimization and data mining in medicine

Wed 07 Apr 2021 13:33:24 AEST

Global optimization : Cutting angle method

Wed 07 Apr 2021 13:33:20 AEST

Nonsmooth optimization approach to clustering

Wed 07 Apr 2021 13:33:20 AEST

2003007532

Derivative-free methods for non-smooth optimization

Wed 07 Apr 2021 13:33:20 AEST

2003007530

Continuous approximations to subdifferentials

Wed 07 Apr 2021 13:33:19 AEST

A multidimensional descent method for global optimization

Wed 07 Apr 2021 13:33:11 AEST

This article presents a new multidimensional descent method for solving global optimization problems with box-constraints. This is a hybrid method where local search method is used for a local descent and global search is used for further multidimensional search on the subsets of intersection of cones generated by the local search method and the feasible region. The discrete gradient method is used for local search and the cutting angle method is used for global search. Two-and three-dimensional cones are used for the global search. Such an approach allows one, as a rule, to escape local minimizers which are not global ones. The proposed method is local optimization method with strong global search properties. We present results of numerical experiments using both smooth and non-smooth global optimization test problems. These results demonstrate that the proposed algorithm allows one to find a global or a near global minimizer.

Special issue of Optimization, dedicated to the development of the ideas of the late Prof. A. Rubinov

Wed 07 Apr 2021 13:33:11 AEST

Optimization methods and the k-committees algorithm for clustering of sequence data

Wed 07 Apr 2021 13:33:11 AEST

The present paper is devoted to new algorithms for unsupervised clustering based on the optimization approaches due to [2], [3] and [4]. We consider a novel situation, where the datasets consist of nucleotide or protein sequences and rather sophisticated biologically significant alignment scores have to be used as a measure of distance. Sequences of this kind cannot be regarded as points in a finite dimensional space. Besides, the alignment scores do not satisfy properties of Minkowski metrics. Nevertheless the optimization approaches have made it possible to introduce a new k-committees algorithm and compare its performance with previous algorithms for two datasets. Our experimental results show that the k-committees algorithms achieves intermediate accuracy for a dataset of ITS sequences, and it can perform better than the discrete k-means and Nearest Neighbour algorithms for certain datasets. All three algorithms achieve good agreement with clusters published in the biological literature before and can be used to obtain biologically significant clusterings.

Fusion strategies for neural learning algorithms using evolutionary and discrete gradient approaches

Wed 07 Apr 2021 13:32:25 AEST

In this paper we investigate different variants for hybrid models using the Discrete Gradient method and an evolutionary strategy for determining the weights in a feed forward artificial neural network. The Discrete Gradient method has the advantage of being able to jump over many local minima and find very deep local minima. However, earlier research has shown that a good starting point for the discrete gradient method can improve the quality of the solution point. Evolutionary algorithms are best suited for global optimisation problems. Nevertheless they are cursed with longer training times and often unsuitable for real world application. For optimisation problems such as weight optimisation for ANNs in real world applications the dimensions are large and time complexity is critical. Hence the idea of a hybrid model can be a suitable option. In this paper we propose different fusion strategies for hybrid models combining the evolutionary strategy with the discrete gradient method to obtain an optimal solution much quicker. Three different fusion strategies are discussed: a linear hybrid model, an iterative hybrid model and a restricted local search hybrid model Comparative results on a range of standard datasets are provided for different fusion hybrid models.

A hybrid neural learning algorithm combining evolutionary algorithm with discrete gradient method

Wed 07 Apr 2021 13:32:24 AEST

Solving Euclidian travelling salesman problem using discrete-gradient based clustering and kohonen neural network

Wed 07 Apr 2021 13:32:24 AEST

Methods and applications of clusterwise linear regression : a survey and comparison

Tue 22 Aug 2023 11:55:58 AEST

Clusterwise linear regression (CLR) is a well-known technique for approximating a data using more than one linear function. It is based on the combination of clustering and multiple linear regression methods. This article provides a comprehensive survey and comparative assessments of CLR including model formulations, description of algorithms, and their performance on small to large-scale synthetic and real-world datasets. Some applications of the CLR algorithms and possible future research directions are also discussed. © 2023 Association for Computing Machinery.

An incremental nonsmooth optimization algorithm for clustering using L1 and L∞ norms

Tue 15 Aug 2023 14:58:19 AEST

An algorithm is developed for solving clustering problems with the similarity measure defined using the L1and L∞ norms. It is based on an incremental approach and applies nonsmooth optimization methods to find cluster centers. Computational results on 12 data sets are reported and the proposed algorithm is compared with the X-means algorithm. ©

Limited Memory Bundle Method for Clusterwise Linear Regression

Tue 15 Aug 2023 14:54:28 AEST

A clusterwise linear regression problem consists of finding a number of linear functions each approximating a subset of the given data. In this paper, the limited memory bundle method is modified and combined with the incremental approach to solve this problem using its nonsmooth optimization formulation. The main contribution of the proposed method is to obtain a fast solution time for large-scale clusterwise linear regression problems. The proposed algorithm is tested on small and large real-world data sets and compared with other algorithms for clusterwise linear regression. Numerical results demonstrate that the proposed algorithm is especially efficient in data sets with large numbers of data points and input variables. © 2022, Springer Nature Switzerland AG.

New gene selection algorithm using hypeboxes to improve performance of classifiers

Tue 01 Nov 2022 13:31:24 AEDT

The use of DNA microarray technology allows to measure the expression levels of thousands of genes in one single experiment which makes possible to apply classification techniques to classify tumours. However, the large number of genes and relatively small number of tumours in gene expression datasets may (and in some cases significantly) diminish the accuracy of many classifiers. Therefore, efficient gene selection algorithms are required to identify most informative genes or groups of genes to improve the performance of classifiers. In this paper, a new gene selection algorithm is developed using marginal hyberboxes of genes or groups of genes for each tumour type. Informative genes are defined using overlaps between hyberboxes. The results on six gene expression datasets demonstrate that the proposed algorithm is able to considerably reduce the number of genes and significantly improve the performance of classifiers. © 2020 Inderscience Enterprises Ltd.

Missing value imputation via clusterwise linear regression

Thu 24 Nov 2022 14:08:28 AEDT

In this paper a new method of preprocessing incomplete data is introduced. The method is based on clusterwise linear regression and it combines two well-known approaches for missing value imputation: linear regression and clustering. The idea is to approximate missing values using only those data points that are somewhat similar to the incomplete data point. A similar idea is used also in clustering based imputation methods. Nevertheless, here the linear regression approach is used within each cluster to accurately predict the missing values, and this is done simultaneously to clustering. The proposed method is tested using some synthetic and real-world data sets and compared with other algorithms for missing value imputations. Numerical results demonstrate that the proposed method produces the most accurate imputations in MCAR and MAR data sets with a clear structure and the percentages of missing data no more than 25%

Multi-objective optimisation to manage trade-offs in water quality and quantity of complex water resource system

Thu 24 Aug 2023 12:52:09 AEST

Water of adequate quality and quantity is the key to health and integrity of the environment and fundamental to good water supply. Achieving water quality and quantity objectives can conflict and has become more complicated with challenges like, climate change, growing populations and changed land uses. Therefore, a multi-objective optimisation strategy is required for achieving optimal water quality and quantity outcomes from a water resources system. This study uses a multi-objective optimisation approach to illustrate the trade-offs occurring when water quantity and quality in a reservoir system are optimised. Taylors Lake, part of the Grampians Reservoir System in Western Victoria, Australia was chosen as the case study for this research as it is quite complex and includes many contemporary water resources challenges seen around the world, such as high turbidity and salinity. The objective functions are set in a way to maximise the water quantity available for supply, while minimising the deviation of quality parameters from the accepted limits. The water system is modelled using eWater Source® modelling platform, while optimisation is undertaken using NSGA-II optimisation technique. Daily time step data over a ten-year period was used in this work. Various optimisation runs were performed with different population sizes and generations to seek out the best trade-off curve. The optimisation results indicate trade-offs between salinity, turbidity, and quantity. Key findings for this case study show that through optimisation, stored water never exceeded 19,000 ML even though the storage capacity was 27,000 ML indicating a significant loss of water to improve quality, or alternatively, a potential asset re-design opportunity.

Nonsmooth optimization-based model and algorithm for semisupervised clustering

Thu 22 Feb 2024 10:20:05 AEDT

Using a nonconvex nonsmooth optimization approach, we introduce a model for semisupervised clustering (SSC) with pairwise constraints. In this model, the objective function is represented as a sum of three terms: the first term reflects the clustering error for unlabeled data points, the second term expresses the error for data points with must-link (ML) constraints, and the third term represents the error for data points with cannot-link (CL) constraints. This function is nonconvex and nonsmooth. To find its optimal solutions, we introduce an adaptive SSC (A-SSC) algorithm. This algorithm is based on the combination of the nonsmooth optimization method and an incremental approach, which involves the auxiliary SSC problem. The algorithm constructs clusters incrementally starting from one cluster and gradually adding one cluster center at each iteration. The solutions to the auxiliary SSC problem are utilized as starting points for solving the nonconvex SSC problem. The discrete gradient method (DGM) of nonsmooth optimization is applied to solve the underlying nonsmooth optimization problems. This method does not require subgradient evaluations and uses only function values. The performance of the A-SSC algorithm is evaluated and compared with four benchmarking SSC algorithms on one synthetic and 12 real-world datasets. Results demonstrate that the proposed algorithm outperforms the other four algorithms in identifying compact and well-separated clusters while satisfying most constraints. © 2021 IEEE.

Comparative analysis of genetic algorithm vs. evolutionary algorithm for hybrid models with discrete gradient method for artificial neural network

Thu 20 Jul 2023 16:05:24 AEST

Minimization of pumping costs in water distribution systems using explicit and implicit pump scheduling

Thu 18 Jan 2024 11:12:44 AEDT

The operation of a water distribution system is a complex task which involves scheduling of pumps, regulating water levels of storages, and providing satisfactory water quality to customers at required flow and pressure. Pump scheduling is one of the most important tasks of the operation of a water distribution system as it represents the major part of its operating costs. In this paper, a novel approach for modeling of pump scheduling to minimize energy consumption by pumps is introduced which uses pump's start/end run times. We separate two types of pumps, one is operated based on the water level in a storage and another one is operated based on downstream pressure. For the first type of pumps both the explicit and implicit pump scheduling can be used, whereas the second type pumps can be optimized only using implicit pump scheduling. The problem is formulated as an optimization problem and an algorithm is developed for its solution. The performance of the algorithm is evaluated using a literature test problem applying the hydraulic simulation model EPANet.

A comparative study of unsupervised classification algorithms in multi-sized data sets

Thu 14 Jul 2022 10:47:05 AEST

The ability to mine and extract useful information automatically, from large data sets, is a common concern for organizations, for the last few decades. Over the internet, data is vastly increasing gradually and consequently the capacity to collect and store very large data is significantly increasing. Existing clustering algorithms are not always efficient and accurate in solving clustering problems for large data sets. However, the development of accurate and fast data classification algorithms for very large scale data sets is still a challenge. In this paper, we present an overview of various algorithms and approaches which are recently being used for Clustering of large data and E-document. In this paper, a comparative study of the performance of various algorithms: the global kmeans algorithm (GKM), the multi-start modified global kmeans algorithm (MS-MGKM), the multi-start kmeans algorithm (MS-KM), the difference of convex clustering algorithm (DCA), the clustering algorithm based on the difference of convex representation of the cluster function and non-smooth optimization (DC-L2), is carried out using C++. © 2019 ACM.

A novel optimization approach towards improving separability of clusters

Thu 14 Dec 2023 14:38:07 AEDT

The objective functions in optimization models of the sum-of-squares clustering problem reflect intra-cluster similarity and inter-cluster dissimilarities and in general, optimal values of these functions can be considered as appropriate measures for compactness of clusters. However, the use of the objective function alone may not lead to the finding of separable clusters. To address this shortcoming in existing models for clustering, we develop a new optimization model where the objective function is represented as a sum of two terms reflecting the compactness and separability of clusters. Based on this model we develop a two-phase incremental clustering algorithm. In the first phase, the clustering function is minimized to find compact clusters and in the second phase, a new model is applied to improve the separability of clusters. The Davies–Bouldin cluster validity index is applied as an additional measure to compare the compactness of clusters and silhouette coefficients are used to estimate the separability of clusters. The performance of the proposed algorithm is demonstrated and compared with that of four other algorithms using synthetic and real-world data sets. Numerical results clearly show that in comparison with other algorithms the new algorithm is able to find clusters with better separability and similar compactness. © 2022

Finding compact and well-separated clusters : clustering using silhouette coefficients

Thu 14 Dec 2023 13:40:15 AEDT

Finding compact and well-separated clusters in data sets is a challenging task. Most clustering algorithms try to minimize certain clustering objective functions. These functions usually reflect the intra-cluster similarity and inter-cluster dissimilarity. However, the use of such functions alone may not lead to the finding of well-separated and, in some cases, compact clusters. Therefore additional measures, called cluster validity indices, are used to estimate the true number of well-separated and compact clusters. Some of these indices are well-suited to be included into the optimization model of the clustering problem. Silhouette coefficients are among such indices. In this paper, a new optimization model of the clustering problem is developed where the clustering function is used as an objective and silhouette coefficients are used to formulate constraints. Then an algorithm, called CLUSCO (CLustering Using Silhouette COefficients), is designed to construct clusters incrementally. Three schemes are discussed to reduce the computational complexity of the algorithm. Its performance is evaluated using fourteen real-world data sets and compared with that of three state-of-the-art clustering algorithms. Results show that the CLUSCO is able to compute compact clusters which are significantly better separable in comparison with those obtained by other algorithms. © 2022 Elsevier Ltd

Bundle methods for nonsmooth DC optimization

Thu 06 Oct 2022 10:27:56 AEDT

This chapter is devoted to algorithms for solving nonsmooth unconstrained difference of convex optimization problems. Different types of stationarity conditions are discussed and the relationship between sets of different stationary points (critical, Clarke stationary and inf-stationary) is established. Bundle methods are developed based on a nonconvex piecewise linear model of the objective function and the convergence of these methods is studied. Numerical results are presented to demonstrate the performance of the methods. © Springer Nature Switzerland AG 2020.

Robust piecewise linear L 1-regression via nonsmooth DC optimization

Thu 01 Jun 2023 13:21:55 AEST

Piecewise linear (Formula presented.) -regression problem is formulated as an unconstrained difference of convex (DC) optimization problem and an algorithm for solving this problem is developed. Auxiliary problems are introduced to design an adaptive approach to generate a suitable piecewise linear regression model and starting points for solving the underlying DC optimization problems. The performance of the proposed algorithm as both approximation and prediction tool is evaluated using synthetic and real-world data sets containing outliers. It is also compared with mainstream machine learning regression algorithms using various performance measures. Results demonstrate that the new algorithm is robust to outliers and in general, provides better predictions than the other alternative regression algorithms for most data sets used in the numerical experiments. © 2020 Informa UK Limited, trading as Taylor & Francis Group.

An augmented subgradient method for minimizing nonsmooth DC functions

Mon 13 Dec 2021 10:19:54 AEDT

A method, called an augmented subgradient method, is developed to solve unconstrained nonsmooth difference of convex (DC) optimization problems. At each iteration of this method search directions are found by using several subgradients of the first DC component and one subgradient of the second DC component of the objective function. The developed method applies an Armijo-type line search procedure to find the next iteration point. It is proved that the sequence of points generated by the method converges to a critical point of the unconstrained DC optimization problem. The performance of the method is demonstrated using academic test problems with nonsmooth DC objective functions and its performance is compared with that of two general nonsmooth optimization solvers and five solvers specifically designed for unconstrained DC optimization. Computational results show that the developed method is efficient and robust for solving nonsmooth DC optimization problems. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

SMGKM : an efficient incremental algorithm for clustering document collections

Mon 10 Jul 2023 14:31:03 AEST

Given a large unlabeled document collection, the aim of this paper is to develop an accurate and efficient algorithm for solving the clustering problem over this collection. Document collections typically contain tens or hundreds of thousands of documents, with thousands or tens of thousands of features (i.e., distinct words). Most existing clustering algorithms struggle to find accurate solutions on such large data sets. The proposed algorithm overcomes this difficulty by an incremental approach, incrementing the number of clusters progressively from an initial value of one to a set value. At each iteration, the new candidate cluster is initialized using a partitioning approach which is guaranteed to minimize the objective function. Experiments have been carried out over six, diverse datasets and with different evaluation criteria, showing that the proposed algorithm has outperformed comparable state-of-the-art clustering algorithms in all cases. © 2023, Springer Nature Switzerland AG.

DC programming algorithm for clusterwise linear L1 regression

Mon 07 Feb 2022 15:31:35 AEDT

The aim of this paper is to develop an algorithm for solving the clusterwise linear least absolute deviations regression problem. This problem is formulated as a nonsmooth nonconvex optimization problem, and the objective function is represented as a difference of convex functions. Optimality conditions are derived by using this representation. An algorithm is designed based on the difference of convex representation and an incremental approach. The proposed algorithm is tested using small to large artificial and real-world data sets. © 2017, Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg.

Analysis of water quantity and quality trade-offs to inform selective harvesting of inflows in complex water resource systems

Mon 01 Aug 2022 11:37:52 AEST

Challenges faced by water resource systems are multi-faceted. The problem can be even more pronounced in a dry continent like Australia where the water resources can often be afflicted by high salinity and turbidity. Therefore, modern water resource systems require to appropriately manage both water quality and quantity. This study aims to illustrate the trade-offs between water quantity and quality in a reservoir, based on decisions to harvest different inflow sources. Taylors Lake of the Grampians reservoir system in Western Victoria, Australia was chosen as the case study for this research as it is sufficiently complex and includes many of the contemporary water resources challenges seen around the world. Different operational scenarios were analysed which included increasingly stringent water quality criteria before the water was harvested or otherwise allowed to by-pass the storage. The study suggests that selective harvesting of water can be an option to improve the overall and long-term water quality within a reservoir, but stringent water quality measures can lead to an associated loss of overall water quantity. This research study provides useful insight to water planners and stakeholders in similar catchment settings around the world, to identify water harvesting regimes with competing water quality constraints. © 2021, The Author(s), under exclusive licence to Springer Nature B.V. Correction to: Analysis of Water Quantity and Quality Trade‑Offs to Inform Selective Harvesting of Inflows in Complex Water Resource Systems (Water Resources Management, (2021), 35, 12, (4149-4165), 10.1007/s11269-021-02936-x)

Subgradient smoothing method for nonsmooth nonconvex optimization

Mon 01 Aug 2022 11:04:08 AEST

In this chapter an unconstrained nonsmooth nonconvex optimization problem is considered and a method for solving this problem is developed. In this method the subproblem for finding search directions is reduced to the unconstrained minimization of a smooth function. This is achieved by using subgradients computed in some neighborhood of a current iteration point and by formulating the search direction finding problem to the minimization of the convex piecewise linear function over the unit ball. The hyperbolic smoothing technique is applied to approximate the minimization problem by a sequence of smooth problems. The convergence of the proposed method is studied and its performance is evaluated using a set of nonsmooth optimization academic test problems. In addition, the method is implemented in GAMS and numerical results using different solvers from GAMS are reported. The proposed method is compared with a number of nonsmooth optimization methods. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.