We investigate Difference of Convex (DC) constrained optimization problems where both the objective function and the constraints are DC and nonsmooth. The problem has applications in a variety of fields, including quadratic programs with complementarity constraints and classification in Machine Learning. We introduce the Constrained Descent-Ascent DC algorithm (CDADC) which generalizes the standard Descent-Ascent framework to incorporate DC constraints using a piecewise affine approximation strategy. The algorithm utilizes the bundle technique to construct models for both the objective and constraint functions. It solves a sequence of convex quadratic subproblems designed to balance objective improvement, proximity to the current iterate, and constraint fulfilment. CDADC avoids evaluation of the objective function’s concave part and, to minimize computational effort, implements bundle resetting whenever a serious step is achieved. We demonstrate finiteness and convergence of the algorithm to a point that satisfies a criterion associated with the B-stationarity notion. The algorithm’s behaviour is illustrated through a couple of numerical examples

Minmaxmin problems are well suited for representing some significant decision making problems, where both strategic and tactical decisions are to be made, at different points of time, in the presence of uncertain scenarios. We survey some basic properties and introduce some classes of structured minmaxmin problems. The main focus is on linear and bilinear minmaxmin problems, which reduce to classic nonsmooth optimization problems. Moreover, two classes of examples are introduced to highlight the practical role of such formulations. The first one is related to the optimal capacity planning of a production-distribution system, and the second one deals with product pricing and distribution in a profit-maximization framework. Finally, focusing on the capacity-planning and product-distribution problem, a computational study has been carried out to illustrate the practical performance of a cutting-plane and proximity-control algorithm for solving the resulting convex nonsmooth minmaxmin model. The numerical results confirm the robustness of the approach and its scalability with respect to both the network size and the number of scenarios.

In supervised classification models, such as Support Vector Machine, the main purpose is to predict the class membership of the incoming samples. In some real applications malicious inputs are inserted to mislead a vulnerable classifier, leading to a wrong prediction. In our work we focus first on the problem of introducing the smallest perturbation of a sample to induce incorrect classification and then on how to produce a significant downgrading of the classifier acting on a subset of the input samples. The novelty of the proposed approach is in the attempt of calculating sparse perturbations by minimizing the relative ℓ _0 -pseudo-norm, which gives rise to a Difference of Convex (DC) optimization model. We present the results of some preliminary experiments.

We present two models in adversarial machine learning, focussing on the Support Vector Machine framework. In particular, we consider both an evasion and a poisoning problem. The first model is aimed at constructing effective sparse perturbation of the dataset samples, while the objective of the second is to induce a substantial rotation of the hyperplane defining the classifier. The two models are formulated as Difference of Convex nonsmooth optimization problems. Numerical results on both synthetic and real life datasets are reported.

We introduce a heuristic rule for calculating the stepsize in the subgradient method for unconstrained convex nonsmooth optimization which, unlike the classic approach, is based on retaining some information from previous iteration. The rule is inspired by the well known two-point stepsize by Barzilai and Borwein (BB) [6] for smooth optimization and it coincides with (BB) in case the function to be minimised is convex quadratic.Under the use of appropriate safeguards we demonstrate that the method terminates at a point that satisfies an approximate optimality condition.The proposed approach is tested in the framework of Lagrangian relaxation for integer linear programming where the Lagrangian dual requires maximization of a concave and nonsmooth (piecewise affine) function. In particular we focus on the relaxation of the Minimum Spanning Tree problem with Conflicting Edge Pairs (MSTC). Comparison with classic subgradient method is presented. The results on some widely used academic test problems are provided too.

We introduce a bundle method for the unconstrained minimization of nonsmooth difference-of-convex (DC) functions, and it is based on the calculation of a special type of descent direction called descent–ascent direction. The algorithm only requires evaluations of the minuend component function at each iterate, and it can be considered as a parsimonious bundle method as accumulation of information takes place only in case the descent–ascent direction does not provide a sufficient decrease. No line search is performed, and proximity control is pursued independent of whether the decrease in the objective function is achieved. Termination of the algorithm at a point satisfying a weak criticality condition is proved, and numerical results on a set of benchmark DC problems are reported. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms – Continuous. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0142 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0142 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

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.

In this paper we propose an heuristic to improve the performances of the recently proposed derivative-free method for nonsmooth optimization CS-DFN. The heuristic is based on a clustering-type technique to compute an estimate of Clarke's generalized gradient of the objective function, obtained via calculation of the (approximate) directional derivative along a certain set of directions. A search direction is then calculated by applying a nonsmooth Newton-type approach. As such, this direction (as it is shown by the numerical experiments) is a good descent direction for the objective function. We report some numerical results and comparison with the original CS-DFN method to show the utility of the proposed improvement on a set of well-known test problems.

Sparse optimization is about finding minimizers of functions characterized by a number of nonzero components as small as possible, such paradigm being of great practical relevance in Machine Learning, particularly in classification approaches based on support vector machines. By exploiting some properties of the k-norm of a vector, namely, of the sum of its k largest absolute-value components, we formulate a sparse optimization problem as a mixed-integer nonlinear program, whose continuous relaxation is equivalent to the unconstrained minimization of a difference-of-convex function. The approach is applied to Feature Selection in the support vector machine framework, and tested on a set of benchmark instances. Numerical comparisons against both the standard ℓ _1 -based support vector machine and a simple version of the Slope method are presented, that demonstrate the effectiveness of our approach in achieving high sparsity level of the solutions without impairing test-correctness.

We propose a novel Mixed-Integer Nonlinear Programming (MINLP) model for sparse optimization based on the polyhedral k-norm. We put special emphasis on the application of sparse optimization in Feature Selection for Support Vector Machine (SVM) classification. We address the continuous relaxation of the problem, which comes out in the form of a fractional programming problem (FPP). In particular, we consider a possible way for tackling FPP by reformulating it via a DC (Difference of Convex) decomposition. We also overview the SVM models and the related Feature Selection in terms of multi-objective optimization. The results of some numerical experiments on benchmark classification datasets are reported.

This repository contains the code to run the experiments present in "The descent-ascent algorithm for DC programming", a bundle method for the unconstrained minimization of nonsmooth difference-of-convex functions.

We tackle the sparsity constrained optimization problem by resorting to polyhedral k-norm as a valid tool to emulate the & ell;0 & ell;0-pseudo-norm. The main novelty of the approach is the use of the dual of the k-norm, which allows to obtain a formulation amenable for a relaxation that can be efficiently handled by block coordinate methods. The advantage of the approach is that it does not require the solution of difference-of-convex programmes, unlike other k-norm based methods available in the literature. In fact, our block coordinate approach requires, at each iteration, the solution of two convex programmes, one of which can be solved in O(n log n)time. We apply the method to feature selection within the framework of Support Vector Machine classification, and we report the results obtained on some benchmark test problems.

We consider, and study with elementary calculus, the polyhedral norms parallel to x parallel to((k)) = sum of the \itk largest among the vertical bar x(i)vertical bar's. Besides their basic properties, we provide various expressions of the unit balls associated with them and determine all the facets and vertices of these balls. We do the same with the dual norm parallel to center dot parallel to((k)) of parallel to center dot parallel to((k)). The study of these polyhedral norms is motivated, among other reasons, by the necessity of handling sparsity in some modern optimization problems, as is explained at the end of the paper.

We present a fast heuristic approach for solving a binary multiple instance learning (MIL) problem, which consists in discriminating between two kinds of item sets: the sets are called bags and the items inside them are called instances. Assuming that only two classes of instances are allowed, a common standard hypothesis states that a bag is positive if it contains at least a positive instance and it is negative when all its instances are negative. Our approach constructs a MIL separating hyperplane by preliminary fixing the normal and reducing the learning phase to a univariate nonsmooth optimization problem, which can be quickly solved by simply exploring the kink points. Numerical results are presented on a set of test problems drawn from the literature.

We present a new disaggregated formulation of the Capacitated Concentrator Location Problem (CCLP) using the notion of cardinality of terminals assigned to a concentrator. This formulation consists of O(mnn) variables and constraints, where m denotes the number of concentrators and n the number of terminals, respectively. We prove that this extended formulation is stronger than the traditional one. We also present two classes of inequalities exploiting the cardinality effect of the extended formulation. The first class is a generalization of the well-known Cover and (1, k)-Configuration inequalities, which collectively are stronger than the original Cover and (1, k)-Configuration inequalities. The second class, called the 2-Facility Cardinality Matching Inequality, holds for the uncapacitated version of the Concentrator Location Problem and can be lifted to become a strong inequality for CCLP. We solve the LP relaxation of the extended formulation and use separation heuristics to identify and sequentially add the previous valid inequalities to improve the lower bound. This approach is embedded in a branch-and-bound and results in a branch-and-cut approach. We test our solution approach on a large set of benchmark problems. The experimentation shows that we can identify the optimal solution at the root node in most of the problem instances with up to 50 concentrators and 50 terminals. For larger sized test problems with up to 100 concentrators and 1000 terminals, the branch-and-cut procedure using the disaggregated formulation outperforms the branch-and-cut procedure applied to the traditional formulation both in terms of CPU and the required number of branch-and-bound nodes. (C) 2019 Elsevier B.V. All rights reserved.

This article addresses the minimum spanning tree problem with conflicting edge pairs, a variant of the classical minimum spanning tree where, given a list of conflicting edges, the goal is to find the cheapest spanning tree with no edges in conflict. We adopt a Lagrangian relaxation approach together with a dual ascent and a subgradient procedure to find tight lower bounds on the optimal solution. The algorithm is also equipped with a heuristic approach which provides an upper bound by removing the conflicts from possible infeasible solutions met during the calculation of the lower bounds. The computational results, carried out on benchmark instances, show that the proposed algorithm finds the optimal solutions on several instances. Moreover, the lower bounds it provides are much more accurate than ones provided by other Lagrangian approaches available in the literature and they are computed in much less time.

We consider the directional sensor network lifetime maximization problem (DSLMP). Given a set of directional sensor and target locations, the problem consists in assigning, at each time unit of a given time horizon, the action radius, the aperture angle, and the orientation direction to all sensors. The objective is to maximize the number of time units when all targets are covered, under certain constraints on sensor available energy. We present a mixed integer nonlinear programming formulation and tackle it by Lagrangean decomposition and subgradient optimization. The algorithm is equipped with a repairing heuristics aimed at finding good‐quality feasible solutions to DSLMP. The results of the application of the proposed approach to a number of problem instances are also reported.

The paper describes a Lagrangian heuristic algorithm for a cross-docking problem, where given amounts of several products must be directly transshipped from a given set of inbound trucks to a given set of out-bound trucks. The cross-docking centre is equipped with some inbound and outbound doors (or gates), where the discharging/loading activities, which are assumed to require a constant time for each truck, take place. The objective is to schedule the activities and to design the transshipment plan, while mini-mizing the ending time of the whole process. Moving along a research line recently traced for the single-door case, the main contribution of the paper is the Lagrangian decomposition scheme for the struc-tured integer linear model of the problem. In particular, decomposition in three subproblems is provided. For all such problems effective solution algorithms are proposed. Two repairing heuristics are embedded into the algorithm for tackling the Lagrangian dual, thus allowing calculation of both lower and upper bounds on the optimal objective function. The performance of the algorithm is evaluated through exten-sive computational experiments on instances of different typologies in terms of number of gates, trucks and products. (c) 2020 Elsevier Ltd. All rights reserved.

We consider polyhedral separation of sets as a possible tool in supervised classification. In particular, we focus on the optimization model introduced by Astorino and Gaudioso (J Optim Theory Appl 112(2):265–293, 2002) and adopt its reformulation in difference of convex (DC) form. We tackle the problem by adapting the algorithm for DC programming known as DCA. We present the results of the implementation of DCA on a number of benchmark classification datasets.