Description:
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.
Description:
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.
Description:
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.
Description:
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.
Description:
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.
Description:
The problem of the estimation of a regression function by continuous piecewise linear functions is formulated as a nonconvex, nonsmooth optimization problem. Estimates are defined by minimization of the empirical L 2 risk over a class of functions, which are defined as maxima of minima of linear functions. An algorithm for finding continuous piecewise linear functions is presented. We observe that the objective function in the optimization problem is semismooth, quasidifferentiable and piecewise partially separable. The use of these properties allow us to design an efficient algorithm for approximation of subgradients of the objective function and to apply the discrete gradient method for its minimization. We present computational results with some simulated data and compare the new estimator with a number of existing ones.
Description:
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, a new version of the k-means algorithm, the global k-means algorithm has been developed. It is an incremental algorithm that dynamically adds one cluster center at a time and uses each data point as a candidate for the k-th cluster center. Results of numerical experiments show that the global k-means algorithm considerably outperforms the k-means algorithms. In this paper, a new version of the global k-means algorithm is proposed. A starting point for the k-th cluster center in this algorithm is computed by minimizing an auxiliary cluster function. Results of numerical experiments on 14 data sets demonstrate the superiority of the new algorithm, however, it requires more computational time than the global k-means algorithm.
Description:
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, a new version of the k-means algorithm, the global k-means algorithm has been developed. It is an incremental algorithm that dynamically adds one cluster center at a time and uses each data point as a candidate for the k-th cluster center. Results of numerical experiments show that the global k-means algorithm considerably outperforms the k-means algorithms. In this paper, a new version of the global k-means algorithm is proposed. A starting point for the k-th cluster center in this algorithm is computed by minimizing an auxiliary cluster function. Results of numerical experiments on 14 data sets demonstrate the superiority of the new algorithm, however, it requires more computational time than the global k-means algorithm.
Description:
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.
Description:
In this paper the problem of localization of wireless sensor network is formulated as an unconstrained nonsmooth optimization problem. We minimize a distance objective function which incorporates unknown sensor nodes and nodes with known positions (anchors) in contrast to popular semidefinite programming (SDP) methods which use artificial objective functions. We study the main properties of the objective function in this problem and design an algorithm for its minimization. Our algorithm is a derivative-free discrete gradient method that allows one to find a near global solution. The algorithm can handle a large number of sensors in the network. This paper contains the theory of our proposed formulation and algorithm while experimental results are included in later work.
Description:
In this paper the problem of localization of wireless sensor network is formulated as an unconstrained nonsmooth optimization problem. We minimize a distance objective function which incorporates unknown sensor nodes and nodes with known positions (anchors) in contrast to popular semidefinite programming (SDP) methods which use artificial objective functions. We study the main properties of the objective function in this problem and design an algorithm for its minimization. Our algorithm is a derivative-free discrete gradient method that allows one to find a near global solution. The algorithm can handle a large number of sensors in the network. This paper contains the theory of our proposed formulation and algorithm while experimental results are included in later work.
Description:
Many optimization problems related to integrated oil and gas production systems are nonconvex and multimodal. Additionally, apart from the innate nonsmoothness of many optimization problems, nonsmooth functions such as minimum and maximum functions may be used to model flow/pressure controllers and cascade mass in the gas gathering and blending networks. In this paper we study the application of different versions of the derivative free Discrete Gradient Method (DGM) as well as the Lipschitz Global Optimizer (LGO) suite to production optimization in integrated oil and gas production systems and their comparison with various local and global solvers used with the General Algebraic Modeling System (GAMS). Four nonconvex and nonsmooth test cases were constructed from a small but realistic integrated gas production system optimization problem. The derivation of the system of equations for the various test cases is also presented. Results demonstrate that DGM is especially effective for solving nonsmooth optimization problems and its two versions are capable global optimization algorithms. We also demonstrate that LGO solves successfully the presented test (as well as other related real-world) problems.
Description:
Many optimization problems related to integrated oil and gas production systems are nonconvex and multimodal. Additionally, apart from the innate nonsmoothness of many optimization problems, nonsmooth functions such as minimum and maximum functions may be used to model flow/pressure controllers and cascade mass in the gas gathering and blending networks. In this paper we study the application of different versions of the derivative free Discrete Gradient Method (DGM) as well as the Lipschitz Global Optimizer (LGO) suite to production optimization in integrated oil and gas production systems and their comparison with various local and global solvers used with the General Algebraic Modeling System (GAMS). Four nonconvex and nonsmooth test cases were constructed from a small but realistic integrated gas production system optimization problem. The derivation of the system of equations for the various test cases is also presented. Results demonstrate that DGM is especially effective for solving nonsmooth optimization problems and its two versions are capable global optimization algorithms. We also demonstrate that LGO solves successfully the presented test (as well as other related real-world) problems.
Description:
The relatively new field of stream mining has necessitated the development of robust drift-aware algorithms that provide accurate, real time, data handling capabilities. Tools are needed to assess and diagnose important trends and investigate drift evolution parameters. In this paper, we present two new and novel visualisation techniques, Pixie and Luna graphs, which incorporate salient group statistics coupled with intuitive visual representations of multidimensional groupings over time. Through the novel representations presented here, spatial interactions between temporal divisions can be diagnosed and overall distribution patterns identified. It provides a means of evaluating in non-constrained capacity, commonly constrained evolutionary problems.
Description:
The relatively new field of stream mining has necessitated the development of robust drift-aware algorithms that provide accurate, real time, data handling capabilities. Tools are needed to assess and diagnose important trends and investigate drift evolution parameters. In this paper, we present two new and novel visualisation techniques, Pixie and Luna graphs, which incorporate salient group statistics coupled with intuitive visual representations of multidimensional groupings over time. Through the novel representations presented here, spatial interactions between temporal divisions can be diagnosed and overall distribution patterns identified. It provides a means of evaluating in non-constrained capacity, commonly constrained evolutionary problems.