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In clusterwise linear regression (CLR), the aim is to simultaneously partition data into a given number of clusters and to find regression coefficients for each cluster. In this paper, we propose a novel approach to model and solve the CLR problem. The main idea is to utilize the support vector machine (SVM) approach to model the CLR problem by using the SVM for regression to approximate each cluster. This new formulation of the CLR problem is represented as an unconstrained nonsmooth optimization problem, where we minimize a difference of two convex (DC) functions. To solve this problem, a method based on the combination of the incremental algorithm and the double bundle method for DC optimization is designed. Numerical experiments are performed to validate the reliability of the new formulation for CLR and the efficiency of the proposed method. The results show that the SVM approach is suitable for solving CLR problems, especially, when there are outliers in data.

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. © 2020, JOURNAL OF INDUSTRIAL AND JOURNAL OF INDUSTRIAL AND, Chongqing Normal University, CNU Australian Government Australian Research Council, ARC: DP190100580, The authors would like to thank two anonymous referees for their comments that helped to improve the quality of the paper. This research by Dr. A. Bagirov was funded by the Australian Government through the Australian Research Council?s Discovery Projects scheme (Project No.: DP190100580). The paper was completed when Dr. A. Bagirov visited Chongqing Normal University, China., Acknowledgements. The authors would like to thank two anonymous referees for their comments that helped to improve the quality of the paper. This research by Dr. A. Bagirov was funded by the Australian Government through the Australian Research Council’s Discovery Projects scheme (Project No.: DP190100580). The paper was completed when Dr. A. Bagirov visited Chongqing Normal University, China.

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.

Intrusion detection systems (IDSs) are devices or software applications that monitor networks or systems for malicious activities and signals alerts/alarms when such activity is discovered. However, an IDS may generate many false alerts which affect its accuracy. In this paper, we develop a cyberattack triage algorithm to detect these alerts (so-called outliers). The proposed algorithm is designed using the clustering, optimization and distance-based approaches. An optimization-based incremental clustering algorithm is proposed to find clusters of different types of cyberattacks. Using a special procedure, a set of clusters is divided into two subsets: normal and stable clusters. Then, outliers are found among stable clusters using an average distance between centroids of normal clusters. The proposed algorithm is evaluated using the well-known IDS data sets—Knowledge Discovery and Data mining Cup 1999 and UNSW-NB15—and compared with some other existing algorithms. Results show that the proposed algorithm has a high detection accuracy and its false negative rate is very low.

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.

Real-time detection of cyber threats is a challenging task in cyber security. With the advancement of technology and ease of access to the internet, more and more individuals and organizations are becoming the target for various cyber attacks such as malware, ransomware, spyware. The target of these attacks is to steal money or valuable information from the victims. Signature-based detection methods fail to keep up with the constantly evolving new threats. Machine learning based detection has drawn more attention of researchers due to its capability of detecting new and modified attacks based on previous attack’s behaviour. The number of malicious activities in a certain domain is significantly low compared to the number of normal activities. Therefore, cyber threats detection data sets are imbalanced. In this paper, we proposed a partial undersampling method to deal with imbalanced data for detecting cyber threats.

In the classical team orienteering problem (TOP), a fixed fleet of vehicles is employed, each of them with a limited driving range. The manager has to decide about the subset of customers to visit, as well as the visiting order (routes). Each customer offers a different reward, which is gathered the first time that it is visited. The goal is then to maximize the total reward collected without exceeding the driving range constraint. This paper analyzes a more realistic version of the TOP in which the driving range limitation is considered as a soft constraint: every time that this range is exceeded, a penalty cost is triggered. This cost is modeled as a piece-wise function, which depends on factors such as the distance of the vehicle to the destination depot. As a result, the traditional reward-maximization objective becomes a non-smooth function. In addition, a second objective, regarding the design of balanced routing plans, is considered as well. A mathematical model for this non-smooth and bi-objective TOP is provided, and a biased-randomized algorithm is proposed as a solving approach.

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.

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.

The aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth optimization, where the objective function is presented as a difference of two convex (DC) func...

Dynamic analysis and pattern matching techniques are widely used in industry, and they provide a straightforward method for the identification of malware samples. Yara is a pattern matching technique that can use sandbox memory dumps for the identification of malware families. However, pattern matching techniques fail silently due to minor code variations, leading to unidentified malware samples. This paper presents a two-layered Malware Variant Identification using Incremental Clustering (MVIIC) process and proposes clustering of unidentified malware samples to enable the identification of malware variants and new malware families. The novel incremental clustering algorithm is used in the identification of new malware variants from the unidentified malware samples. This research shows that clustering can provide a higher level of performance than Yara rules, and that clustering is resistant to small changes introduced by malware variants. This paper proposes a hybrid approach, using Yara scanning to eliminate known malware, followed by clustering, acting in concert, to allow the identification of new malware variants. F1 score and V-Measure clustering metrics are used to evaluate our results.

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.

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.

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 $$\varepsilon $$ź-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.

The Internet of Things (IoT) has significantly increased the number of devices connected to the Internet ranging from sensors to multi-source data information. As the IoT continues to evolve with new technologies number of threats and attacks against IoT devices are on the increase. Analyzing and detecting these attacks originating from different sources needs machine learning models. These models provide proactive solutions for detecting attacks and their sources. In this paper, we propose to apply a supervised machine learning classification technique to identify cyber-attacks from each source. More precisely, we apply the incremental piecewise linear classifier that constructs boundary between sources/classes incrementally starting with one hyperplane and adding more hyperplanes at each iteration. The algorithm terminates when no further significant improvement of the separation of sources/classes is possible. The construction and usage of piecewise linear boundaries allows us to avoid any possible overfitting. We apply the incremental piecewise linear classifier on the multi-source real world cyber security data set to identify cyber-attacks and their sources.

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%

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.

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 non-drinking 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.

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.

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.

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. (C) 2013 Elsevier B.V. All rights reserved.

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.

Impact of water quality conditions in sources on the optimal operation of a regional multiquality water distribution system is analysed. Three operational objectives are concurrently minimised, being pump energy costs, turbidity and salinity deviations at customer nodes. The optimisation problem is solved using GANetXL (NSGA-II) linked with EPANet. The example network incorporates scenarios with different water quality in sources. It was discovered that two types of tradeoffs, competing and non-competing, exist between the objectives and that the type of tradeoff is not unique between a particular pair of objectives across scenarios. The findings may be used for system operational planning.

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 Spath algorithm on several publicly available data sets for regression analysis.

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.

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.

This volume of the Journal of Global Optimization is devoted to papers presented at the 25th European Conference on Operational Research, EURO XXV 2012, which was held on July 811, 2012, in Vilnius, Lithuania. The conference attracted 2044 registered participants from 68 countries fromall the continents across theORcommunity. ContinuousOptimizationwas one of the largest main areas in the entire conferencewith about 250 participants. They exchanged experiences in solving real-world problems, discussed recent achievements in optimization theory, methods and applications, reported on developments and implementations of appropriate models and efficient solution procedures for problems of continuous optimization. The EURO XXV 2012 conference provided an excellent forum for researchers and practitioners to promote their recent advances in continuous optimization to the wider scientific community, to identify new research challenges as well as promising research developments in theory, methods and applications and to promote interactions with colleagues from related research areas of modern OR and its emerging applications. In this spirit, this special issue is devoted to the theme connecting sciences supported by global optimization. For this special issue, participants of EURO XXV 2012 were invited to submit papers on continuous optimization and related topics. The papers included recent theoretical and applied contributions in various fields including linear, nonlinear, stochastic, parametric and dynamic optimization as well as control theory. Based on rigorous reviewing processes, six papers

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.

An L 2-boosting algorithm for estimation of a regression function from random design is presented, which consists of fitting repeatedly a function from a fixed nonlinear function space to the residuals of the data by least squares and by defining the estimate as a linear combination of the resulting least squares estimates. Splitting of the sample is used to decide after how many iterations of smoothing of the residuals the algorithm terminates. The rate of convergence of the algorithm is analyzed in case of an unbounded response variable. The method is used to fit a sum of maxima of minima of linear functions to a given data set, and is compared with other nonparametric regression estimates using simulated data.

In this paper, estimation of a regression function from independent and identically distributed random variables is considered. Estimates are defined by minimization of the empirical L2 risk over a class of functions, which are defined as maxima of minima of linear functions. Results concerning the rate of convergence of the estimates are derived. In particular, it is shown that for smooth regression functions satisfying the assumption of single index models, the estimate is able to achieve (up to some logarithmic factor) the corresponding optimal one-dimensional rate of convergence. Hence, under these assumptions, the estimate is able to circumvent the so-called curse of dimensionality. The small sample behavior of the estimates is illustrated by applying them to simulated data.

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.

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.

A new derivative-free method is developed for solving unconstrained nonsmooth optimization problems. This method is based on the notion of a discrete gradient. It is demonstrated that the discrete gradients can be used to approximate subgradients of a broad class of nonsmooth functions. It is also shown that the discrete gradients can be applied to find descent directions of nonsmooth functions. The preliminary results of numerical experiments with unconstrained nonsmooth optimization problems as well as the comparison of the proposed method with the nonsmooth optimization solver DNLP from CONOPT-GAMS and the derivative-free optimization solver CONDOR are presented.

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.

Missing values may be present in data without undermining its use for diagnostic / classification purposes but compromise applicationof readily available software. Surrogate entries can remedy the situation, although the outcome is generally unknown.Discretization of continuous attributes renders all data nominal and is helpful in dealing with missing values; particularly, nospecial handling is required for different attribute types. A number of classifiers exist or can be reformulated for this representation.Some classifiers can be reinvented as data completion methods. In this work the Decision Tree, Nearest Neighbour,and Naive Bayesian methods are demonstrated to have the required aptness. An approach is implemented whereby the enteredmissing values are not necessarily a close match of the true data; however, they intend to cause the least hindrance for classification.The proposed techniques find their application particularly in medical diagnostics. Where clinical data represents anumber of related conditions, taking Cartesian product of class values of the underlying sub-problems allows narrowing downof the selection of missing value substitutes. Real-world data examples, some publically available, are enlisted for testing. Theproposed and benchmark methods are compared by classifying the data before and after missing value imputation, indicating asignificant improvement.

CONTINUOUS SUBDIFFERENTIAL APPROXIMATION AND ITS CONSTRUCTION

WOS: 000361624700018, 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., Australian Research Council [DP140103213], The authors would like to thank three anonymous referees for their criticism and comments which allowed to improve the quality of the paper. The research by Dr. A.M. Bagirov was supported under Australian Research Council's Discovery Projects funding scheme (Project No. DP140103213).

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 $$k$$ k -means and modified global $$k$$ 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 $$k$$ k -means algorithm in finding global solutions. We use an auxiliary cluster problem to generate a set of initial points and apply the $$k$$ 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 $$k$$ k -means algorithms in finding global solutions to clustering problems.

We examine numerical performance of various methods of calculation of the Conditional Value-at-risk (CVaR), and portfolio optimization with respect to this risk measure. We concentrate on the method proposed by Rockafellar and Uryasev in (Rockafellar, R.T. and Uryasev, S., 2000, Optimization of conditional value-at-risk. Journal of Risk, 2, 21–41), which converts this problem to that of convex optimization. We compare the use of linear programming techniques against a non-smooth optimization method of the discrete gradient, and establish the supremacy of the latter. We show that non-smooth optimization can be used efficiently for large portfolio optimization, and also examine parallel execution of this method on computer clusters.

This paper introduces the notion of piecewise partially separable functions and studies their properties. We also consider some of many applications of these functions. Finally, we consider the problem of minimizing of piecewise partially separable functions and develop an algorithm for its solution. This algorithm exploits the structure of such functions. We present the results of preliminary numerical experiments.

The minimum sum-of-squares clustering problem is formulated as a problem of nonsmooth, nonconvex optimization, and an algorithm for solving the former problem based on nonsmooth optimization techniques is developed. The issue of applying this algorithm to large data sets is discussed. Results of numerical experiments have been presented which demonstrate the effectiveness of the proposed algorithm.

This paper develops a new derivative-free method for solving linearly constrained nonsmooth optimization problems. The objective functions in these problems are, in general, non-regular locally Lipschitz continuous function. The computation of generalized subgradients of such functions is difficult task. In this paper we suggest an algorithm for the computation of subgradients of a broad class of non-regular locally continuous Lipschitz functions. This algorithm is based on the notion of a discrete gradient. An algorithm for solving linearly constrained nonsmooth optimization problems based on discrete gradients is developed. We report preliminary results of numerical experiments. These results demonstrate that the proposed algorithm is efficient for solving linearly constrained nonsmooth optimization problems.

The problem of cluster analysis is formulated as a problem of nonsmooth, nonconvex optimization. An algorithm for solving the latter optimization problem is developed which allows one to significantly reduce the computational efforts. This algorithm is based on the so-called discrete gradient method. Results of numerical experiments are presented which demonstrate the effectiveness of the proposed algorithm.

This paper presents a new method for solving global optimization problems. We use a local technique based on the notion of discrete gradients for finding a cone of descent directions and then we use a global cutting angle algorithm for finding global minimum within the intersection of the cone and the feasible region. We present results of numerical experiments with well-known test problems and with the so-called cluster function. These results confirm that the proposed algorithms allows one to find a global minimizer or at least a deep local minimizer of a function with a huge amount of shallow local minima.