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40103 Numerical and Computational Mathematics
20802 Computation Theory and Mathematics
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A heuristic algorithm for solving the minimum sum-of-squares clustering problems

**Authors:**Ordin, Burak , Bagirov, Adil**Date:**2015**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 61, no. 2 (2015), p. 341-361**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**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.

- Bagirov, Adil, Barton, Andrew, Mala-Jetmarova, Helena, Al Nuaimat, Alia, Ahmed, S. T., Sultanova, Nargiz, Yearwood, John

**Authors:**Bagirov, Adil , Barton, Andrew , Mala-Jetmarova, Helena , Al Nuaimat, Alia , Ahmed, S. T. , Sultanova, Nargiz , Yearwood, John**Date:**2013**Type:**Text , Journal article**Relation:**Mathematical and Computer Modelling Vol. 57, no. 3-4 (2013), p. 873-886**Relation:**http://purl.org/au-research/grants/arc/LP0990908**Full Text:**false**Reviewed:****Description:**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.**Description:**2003010583

An approximate subgradient algorithm for unconstrained nonsmooth, nonconvex optimization

- Bagirov, Adil, Ganjehlou, Asef Nazari

**Authors:**Bagirov, Adil , Ganjehlou, Asef Nazari**Date:**2008**Type:**Text , Journal article**Relation:**Mathematical Methods of Operations Research Vol. 67, no. 2 (2008), p. 187-206**Relation:**http://purl.org/au-research/grants/arc/DP0666061**Full Text:****Reviewed:****Description:**In this paper a new algorithm for minimizing locally Lipschitz functions is developed. Descent directions in this algorithm are computed by solving a system of linear inequalities. The convergence of the algorithm is proved for quasidifferentiable semismooth functions. We present the results of numerical experiments with both regular and nonregular objective functions. We also compare the proposed algorithm with two different versions of the subgradient method using the results of numerical experiments. These results demonstrate the superiority of the proposed algorithm over the subgradient method. © 2007 Springer-Verlag.**Description:**C1

**Authors:**Bagirov, Adil , Ganjehlou, Asef Nazari**Date:**2008**Type:**Text , Journal article**Relation:**Mathematical Methods of Operations Research Vol. 67, no. 2 (2008), p. 187-206**Relation:**http://purl.org/au-research/grants/arc/DP0666061**Full Text:****Reviewed:****Description:**In this paper a new algorithm for minimizing locally Lipschitz functions is developed. Descent directions in this algorithm are computed by solving a system of linear inequalities. The convergence of the algorithm is proved for quasidifferentiable semismooth functions. We present the results of numerical experiments with both regular and nonregular objective functions. We also compare the proposed algorithm with two different versions of the subgradient method using the results of numerical experiments. These results demonstrate the superiority of the proposed algorithm over the subgradient method. © 2007 Springer-Verlag.**Description:**C1

Comparative study of RPSALG algorithm for convex semi-infinite programming

- Auslender, Alfred, Ferrer, Albert, Goberna, Miguel, López, Marco

**Authors:**Auslender, Alfred , Ferrer, Albert , Goberna, Miguel , López, Marco**Date:**2014**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 60, no. 1 (2014), p. 59-87**Full Text:**false**Reviewed:****Description:**The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convex semi-infinite programing problems, whose convergence was analyzed in a previous paper of three of the authors. In this paper we consider a partial implementation of RPSALG for solving ordinary convex semi-infinite programming problems. Each iteration of RPSALG involves two types of auxiliary optimization problems: the first one consists of obtaining an approximate solution of some discretized convex problem, while the second one requires to solve a non-convex optimization problem involving the parametric constraints as objective function with the parameter as variable. In this paper we tackle the latter problem with a variant of the cutting angle method called ECAM, a global optimization procedure for solving Lipschitz programming problems. We implement different variants of RPSALG which are compared with the unique publicly available SIP solver, NSIPS, on a battery of test problems.

Structure learning of Bayesian Networks using global optimization with applications in data classification

- Taheri, Sona, Mammadov, Musa

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 9, no. 5 (2014), p. 931-948**Full Text:****Reviewed:****Description:**Bayesian Networks are increasingly popular methods of modeling uncertainty in artificial intelligence and machine learning. A Bayesian Network consists of a directed acyclic graph in which each node represents a variable and each arc represents probabilistic dependency between two variables. Constructing a Bayesian Network from data is a learning process that consists of two steps: learning structure and learning parameter. Learning a network structure from data is the most difficult task in this process. This paper presents a new algorithm for constructing an optimal structure for Bayesian Networks based on optimization. The algorithm has two major parts. First, we define an optimization model to find the better network graphs. Then, we apply an optimization approach for removing possible cycles from the directed graphs obtained in the first part which is the first of its kind in the literature. The main advantage of the proposed method is that the maximal number of parents for variables is not fixed a priory and it is defined during the optimization procedure. It also considers all networks including cyclic ones and then choose a best structure by applying a global optimization method. To show the efficiency of the algorithm, several closely related algorithms including unrestricted dependency Bayesian Network algorithm, as well as, benchmarks algorithms SVM and C4.5 are employed for comparison. We apply these algorithms on data classification; data sets are taken from the UCI machine learning repository and the LIBSVM. © 2014, Springer-Verlag Berlin Heidelberg.

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 9, no. 5 (2014), p. 931-948**Full Text:****Reviewed:****Description:**Bayesian Networks are increasingly popular methods of modeling uncertainty in artificial intelligence and machine learning. A Bayesian Network consists of a directed acyclic graph in which each node represents a variable and each arc represents probabilistic dependency between two variables. Constructing a Bayesian Network from data is a learning process that consists of two steps: learning structure and learning parameter. Learning a network structure from data is the most difficult task in this process. This paper presents a new algorithm for constructing an optimal structure for Bayesian Networks based on optimization. The algorithm has two major parts. First, we define an optimization model to find the better network graphs. Then, we apply an optimization approach for removing possible cycles from the directed graphs obtained in the first part which is the first of its kind in the literature. The main advantage of the proposed method is that the maximal number of parents for variables is not fixed a priory and it is defined during the optimization procedure. It also considers all networks including cyclic ones and then choose a best structure by applying a global optimization method. To show the efficiency of the algorithm, several closely related algorithms including unrestricted dependency Bayesian Network algorithm, as well as, benchmarks algorithms SVM and C4.5 are employed for comparison. We apply these algorithms on data classification; data sets are taken from the UCI machine learning repository and the LIBSVM. © 2014, Springer-Verlag Berlin Heidelberg.

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