Special Issue "Nonsmooth Optimization in Honor of the 60th Birthday of Adil M. Bagirov": Foreword by Guest Editors

Nonsmooth optimization refers to the general problem of minimizing (or maximizing) functions that have discontinuous gradients. This Special Issue contains six research articles that collect together the most recent techniques and applications in the area of nonsmooth optimization. These include novel techniques utilizing some decomposable structures in nonsmooth problems-for instance, the difference-of-convex (DC) structure-and interesting important practical problems, like multiple instance learning, hydrothermal unit-commitment problem, and scheduling the disposal of nuclear waste

Splitting Metrics Diagonal Bundle Method for Large-Scale Nonconvex and Nonsmooth Optimization

Nonsmooth optimization is traditionally based on convex analysis and most solution methods rely strongly on the convexity of the problem. In this paper, we propose an efficient diagonal bundle method for nonconvex large-scale nonsmooth optimization. The novelty of the new method is in different usage of metrics depending on the convex or concave behaviour of the objective at the current iteration point. The usage of different metrics gives us a possibility to better deal with the nonconvexity of the problem than the sole — the most commonly used and quite arbitrary — downward shifting of the piecewise linear model does. The convergence of the proposed method is proved for semismooth functions that are not necessary differentiable nor convex. The numerical experiments have been made using problems with up to million variables. The results to be presented confirm the usability of the new method.</p

Bundle enrichment method for nonsmooth difference of convex programming problems

The Bundle Enrichment Method (BEM-DC) is introduced for solving nonsmooth difference of convex (DC) programming problems. The novelty of the method consists of the dynamic management of the bundle. More specifically, a DC model, being the difference of two convex piecewise affine functions, is formulated. The (global) minimization of the model is tackled by solving a set of convex problems whose cardinality depends on the number of linearizations adopted to approximate the second DC component function. The new bundle management policy distributes the information coming from previous iterations to separately model the DC components of the objective function. Such a distribution is driven by the sign of linearization errors. If the displacement suggested by the model minimization provides no sufficient decrease of the objective function, then the temporary enrichment of the cutting plane approximation of just the first DC component function takes place until either the termination of the algorithm is certified or a sufficient decrease is achieved. The convergence of the BEM-DC method is studied, and computational results on a set of academic test problems with nonsmooth DC objective functions are provided. © 2023 by the authors

Clustering in large data sets with the limited memory bundle method

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

Aggregate subgradient method for nonsmooth DC optimization

The aggregate subgradient method is developed for solving unconstrained nonsmooth difference of convex (DC) optimization problems. The proposed method shares some similarities with both the subgradient and the bundle methods. Aggregate subgradients are defined as a convex combination of subgradients computed at null steps between two serious steps. At each iteration search directions are found using only two subgradients: the aggregate subgradient and a subgradient computed at the current null step. It is proved that the proposed method converges to a critical point of the DC optimization problem and also that the number of null steps between two serious steps is finite. The new method is tested using some academic test problems and compared with several other nonsmooth DC optimization solvers. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature

Double bundle method for finding clarke stationary points in nonsmooth dc programming

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) functions. The novelty in our method is a new escape procedure which enables us to guarantee approximate Clarke stationarity for solutions by utilizing the DC components of the objective function. This optimality condition is stronger than the criticality condition typically used in DC programming. Moreover, if a candidate solution is not approximate Clarke stationary, then the escape procedure returns a descent direction. With this escape procedure, we can avoid some shortcomings encountered when criticality is used. The finite termination of the double bundle method to an approximate Clarke stationary point is proved by assuming that the subdifferentials of DC components are polytopes. Finally, some encouraging numerical results are presented

A New Subgradient Based Method for Nonsmooth DC Programming

The aggregate subgradient method is developed for solving unconstrained nonsmooth difference of convex (DC) optimization problems. The proposed method shares some similarities with both the subgradient and the bundle methods. Aggregate subgradients are defined as a convex combination of subgradients computed at null steps between two serious steps. At each iteration search directions are found using only two subgradients: the aggregate subgradient and a subgradient computed at the current null step. It is proved that the proposed method converges to a critical point of the DC optimization problem and also that the number of null steps between two serious steps is finite. The new method is tested using some academic test problems and compared with several other nonsmooth DC optimization solvers.</p

New bundle method for clusterwise linear regression utilizing support vector machines

Clusterwise linear regression (CLR) aims to simultaneously partition a data into a given number of clusters and find regression coefficients for each cluster. In this paper, we propose a novel approach to 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 CLR is represented as an unconstrained nonsmooth optimization problem, where the objective function is a difference of convex (DC) functions. A method based on the combination of the incremental algorithm and the double bundle method for DC optimization is designed to solve it. Numerical experiments are made to validate the reliability of the new formulation and the efficiency of the proposed method. The results show that the SVM approach is beneficial in solving CLR problems, especially, when there are outliers in data.</p

Aggregate subgradient method for nonsmooth DC optimization

The aggregate subgradient method is developed for solving unconstrained nonsmooth difference of convex (DC) optimization problems. The proposed method shares some similarities with both the subgradient and the bundle methods. Aggregate subgradients are defined as a convex combination of subgradients computed at null steps between two serious steps. At each iteration search directions are found using only two subgradients: the aggregate subgradient and a subgradient computed at the current null step. It is proved that the proposed method converges to a critical point of the DC optimization problem and also that the number of null steps between two serious steps is finite. The new method is tested using some academic test problems and compared with several other nonsmooth DC optimization solvers