Glob Optim Research Articles

The horizontal tensor complementarity problem

This article explores a new type of nonlinear complementarity problem, namely the horizontal tensor complementarity problem (HTCP), which is a natural extension of the horizontal linear complementarity problem studied in Sznajder [Degree-theoretic analysis of the vertical and horizontal linear complementarity problems [Ph.D. Thesis]. University of Maryland Baltimore County; 1994]. We extend the concepts of R 0 , R and P pairs from a pair of linear transformations given in Chi et al. [The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra. J Global Optim. 2019;73:153–169] to a pair of tensors. When a given pair of tensors has these properties, we use degree-theoretic tools to discuss the existence and boundedness of solutions to the HTCP. Finally, we study a uniqueness result of the solution of the HTCP.

Enhancements of discretization approaches for non-convex mixed-integer quadratically constrained quadratic programming: Part I

We study mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We present MIP relaxation methods for non-convex continuous variable products. In this paper, we consider MIP relaxations based on separable reformulation. The main focus is the introduction of the enhanced separable MIP relaxation for non-convex quadratic products of the form z=xy\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z=xy$$\\end{document}, called hybrid separable (HybS). Additionally, we introduce a logarithmic MIP relaxation for univariate quadratic terms, called sawtooth relaxation, based on Beach (Beach in J Glob Optim 84:869–912, 2022). We combine the latter with HybS and existing separable reformulations to derive MIP relaxations of MIQCQPs. We provide a comprehensive theoretical analysis of these techniques, underlining the theoretical advantages of HybS compared to its predecessors. We perform a broad computational study to demonstrate the effectiveness of the enhanced MIP relaxation in terms of producing tight dual bounds for MIQCQPs. In Part II, we study MIP relaxations that extend the MIP relaxation normalized multiparametric disaggregation technique (NMDT) (Castro in J Glob Optim 64:765–784, 2015) and present a computational study which also includes the MIP relaxations from this work and compares them with a state-of-the-art of MIQCQP solvers.

On the diagonal proximal point algorithms

We study the asymptotic behaviour of the diagonal proximal point algorithms governed by a sequence of maximally monotone operators. With a summability condition on the Brézis–Haraux functions of the sequence of operators, we obtain several ergodic and weak convergence results in the monotone and subgradient cases. This approach is of relevance to and motivated by Attouch et al. [Asymptotic behaviour of nonautonomous monotone and subgradient evolution equations. Trans Amer Math Soc. 2018;370:755–790]. Some strong convergence results are presented as well. Our results in this paper include discrete versions of the results in Attouch et al. [Asymptotic behaviour of nonautonomous monotone and subgradient evolution equations. Trans Amer Math Soc. 2018;370:755–790], and also unify and improve many existing results in Attouch et al. [Prox-penalization and splitting methods for constrained variational problems. SIAM J Optim. 2011;21:149–173], Cabot [Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization. SIAM J Optim. 2005;15:555–572], Lehdili and Moudafi [Combining the proximal algorithm and Tikhonov regularization. Optimization. 1996;37:239–252], Xu [A regularization method for the proximal point algorithm. J Glob Optim. 2006;36:115–125].

New formulas for subdifferentials of perturbed distance functions

ABSTRACT We give exact formulas for the subdifferentials of perturbed distance functions in a normed space. Our method, seemingly novel and different from existing ones, is to turn the involved problem equivalently to a parametric optimization problem and then apply variational analysis technique to the optimal value function. In the convex setting, we obtain new representations for the subdifferential of perturbed distance functions, which do not depend on the relative position of the reference point with respect to the input set, and which are described directly via the input data. Our results complement those of Wang et al. [J. Global Optim. 2010;46:489–501] and of Li and Bounkhel [Nonlinear Anal. 2014;108:173–188], which were established by different methods.

Computational advances in polynomial optimization: RAPOSa, a freely available global solver

In this paper we introduce RAPOSa, a global optimization solver specifically designed for (continuous) polynomial programming problems with box-constrained variables. Written entirely in C++, RAPOSa is based on the Reformulation-Linearization (Sherali and Tuncbilek in J Glob Optim 103:225–249, 1992). We present a description of the main characteristics of RAPOSa along with a thorough analysis of the impact on its performance of various enhancements discussed in the literature, such as bound tightening and SDP cuts. We also present a comparative study with three of the main state-of-the-art global optimization solvers: BARON, Couenne and SCIP.

Limit sets in global multiobjective optimization

ABSTRACT Inspired by the recently introduced branch-and-bound method for continuous multiobjective optimization problems from G. Eichfelder, P. Kirst, L. Meng, O. Stein [A general branch-and-bound framework for continuous global multiobjective optimization. J Glob Optim. 2021;80:195–227], we study for a general class of branch-and-bound methods in which sense the generated terminal enclosure and the terminal provisional nondominated set approximate the nondominated set when the termination accuracy is driven to zero. Our convergence analysis of the enclosures relies on constructions from the above paper, but is self-contained and also covers the mixed-integer case. The analysis for the provisional nondominated set is based on general convergence properties of the epsilon-nondominated set, and hence it is also applicable to other algorithms which generate such points. Furthermore, we discuss post-processing steps for the terminal enclosure and provide numerical illustrations for the cases of two and three objective functions.

On relations between nonsmooth interval-valued multiobjective programming problems and generalized Stampacchia vector variational inequalities

In this article, we study nonsmooth interval-valued multiobjective programming problem and generalized Stampacchia vector variational inequality with its weak form for interval-valued functions. Using the tools of Mordukhovich subdifferential, we define some new classes of generalized approximate LU-convex functions. These functions are then employed to establish the relations between the solutions of generalized vector variational inequalities and the approximate LU-efficient solutions of the nonsmooth interval-valued multiobjective programming problem. Moreover, we identify the Kuhn–Tucker vector critical points of the considered nonsmooth interval-valued multiobjective programming problem. Under suitable constraint qualification, we establish the equivalence among local approximate LU-efficient points, Kuhn–Tucker vector critical points and the solutions of generalized Stampacchia vector variational inequalities. The results of this paper extend and sharpen the corresponding results of [Giannessi F. On Minty variational principle. In: Giannessi F, Komlósi S, Rapcśak T. editors. New trends in mathematical programming. Dordrecht: Kluwer Academic Publishers; 1997. p. 93–99], [Golestani M, Sadeghi H, Tavan Y. Nonsmooth multiobjective problems and generalized vector variational inequalities using quasi-efficiency. J Optim Theory Appl. 2018;179(3):896–916], [Lee GM, Lee KB. Vector variational inequalities for nondifferentiable convex vector optimization problems. J Global Optim. 2005;32(4):597–612], [Mishra SK, Upadhyay BB. Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency. Positivity. 2013;17(4):1071–1083], [Upadhyay BB, Mohapatra RN, Mishra SK. On relationships between vector variational inequality and nonsmooth vector optimization problems via strict minimizers. Adv Nonlinear Var Inequal. 2017;20(2):1–12], [Zhang J, Zheng Q,Ma X, Li L. Relationships between interval-valued vector optimization problems and vector variational inequalities. Fuzzy Optim Decis Mak. 2016;15(1):33–55] for nonsmooth interval-valued multiobjective programming problem by using the powerful tool of Mordukhovich subdifferential.

Bilevel optimization for joint scheduling of production and energy systems

Energy-intensive production sites are often supplied with energy by on-site energy systems. Commonly, the scheduling of the systems is performed sequentially, starting with the scheduling of the production system. Often, the on-site energy system is operated by a different company than the production system. In consequence, the production and the energy system schedule their operation towards misaligned objectives leading in general to suboptimal schedules for both systems. To reflect the independent optimization with misaligned objectives, the scheduling problem of the production system can be formulated as a bilevel problem. We formulate the bilevel problem with mixed-integer decision variables in the upper and the lower level, and propose an algorithm to solve this bilevel problem based on the deterministic and global algorithm by Djelassi, Glass and Mitsos (J Glob Optim 75:341–392, 2019. https://doi.org/10.1007/s10898-019-00764-3) for bilevel problems with coupling equality constraints. The algorithm works by discretizing the independent lower-level variables. In the scheduling problem considered herein, the only coupling equality constraints are energy balances in the lower level. Since an intuitive distinction is missing between dependent and independent variables, we specialize the algorithm and add a procedure to identify independent variables to be discretized. Thereby, we preserve convergence guarantees. The performance of the algorithm is demonstrated in two case studies. In the case studies, the production system favors different technologies for the energy supply than the energy system. By solving the bilevel problem, the production system identifies an energy demand, which leads to minimal cost. Additionally, we demonstrate the benefits of solving the bilevel problem instead of solving the common integrated or sequential problem.

Optimality conditions for mathematical programs with equilibrium constraints using directional convexificators

We consider a nonsmooth and nonconvex mathematical program with equilibrium constraints (MPEC) where its functions are not necessarily locally Lipschitz/continuous. In exploiting the idea of directional convexificators introduced by Dempe and Pelicka [Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming. J Global Optim. 2015;61:769–788], we define two forms of nonsmooth versions of the classical Abadie constraint qualifications, and introduce nonsmooth stationarity conditions, which are based on directional convexificators. Then, we employ these conditions to derive first-order optimality conditions subject to generalized Abadie constraint qualifications. Finally, we establish sufficient optimality conditions for MPEC under a new assumption of generalized convexity.

On reduction of exhausters via a support function representation

Exhausters are families of compact, convex sets which provide minmax or maxmin representations of positively homogeneous functions and they are efficient tools for the study of nonsmooth functions (Demyanov in Optimization 45:13–29, 1999). Upper and lower exhausters of positively homogeneous functions are employed to describe optimality conditions in geometric terms and also to find directions of steepest descent or ascent. Since an upper/lower exhauster may contain finitely or infinitely many compact convex sets, the problem of minimality and reduction of exhausters naturally arise. There are several approaches to reduce exhausters (Abbasov in J Glob Optim 74(4):737–751, 2019; Grzybowski et al. in J Glob Optim 46(4):589–601, 2010; Küçük et al. in J Optim Theory Appl 165:693–707, 2015; Roshchina in J Optim Theory Appl 136:261–273, 2008; J Convex Anal 15(4):859–868, 2008). In this study, in the sense of inclusion-minimality, some reduction techniques for upper exhausters of positively homogeneous functions defined from \(\mathbb {R}^2\) to \(\mathbb {R}\) is proposed by means of a representation of support functions. These techniques have concrete geometric meanings and they form a basis for a necessary and sufficient condition for inclusion-minimality of exhausters. Some examples are presented to illustrate each reduction technique.

A modified simplex partition algorithm to test copositivity

A real symmetric matrix A is copositive if x^top Axge 0 for all xge 0. As A is copositive if and only if it is copositive on the standard simplex, algorithms to determine copositivity, such as those in Sponsel et al. (J Glob Optim 52:537–551, 2012) and Tanaka and Yoshise (Pac J Optim 11:101–120, 2015), are based upon the creation of increasingly fine simplicial partitions of simplices, testing for copositivity on each. We present a variant that decomposes a simplex bigtriangleup , say with n vertices, into a simplex bigtriangleup _1 and a polyhedron varOmega _1; and then partitions varOmega _1 into a set of at most (n-1) simplices. We show that if A is copositive on varOmega _1 then A is copositive on bigtriangleup _1, allowing us to remove bigtriangleup _1 from further consideration. Numerical results from examples that arise from the maximum clique problem show a significant reduction in the time needed to establish copositivity of matrices.

A Generalized Ky Fan Minimax Inequality on Finite-Dimensional Spaces

An existence result for a generalized inequality over a possible unbounded domain in a finite-dimensional space is established. The proof technique allows to avoid any monotonicity assumption. We adapt a weak coercivity condition introduced in Castellani and Giuli (J Glob Optim 75:163–176, 2019) for a generalized game which extends an older one proposed by Konnov and Dyabilkin (J Glob Optim 49:575–577, 2011) for equilibrium problems. Our main result encompasses and generalizes several existence results for equilibrium, quasiequilibrium and fixed-point problems.

An Efficient Algorithm for the Extended Trust-Region Subproblem with Two Linear Constraints

In this paper, we study the extended trust-region subproblem with two linear inequality constraints (2-eTRS). Without any assumptions on the linear constraints, it is shown that its global optimal solution can be found by solving at most four trust-region subproblems. Finally, the efficiency of the proposed algorithm is compared with the branch and bound algorithm of Beck et al. (J Glob Optim 69(2):309–342, 2017) and CVX software on several classes of test problems.

Necessary conditions for weak minima and for strict minima of order two in nonsmooth constrained multiobjective optimization

In this paper, we give necessary conditions for the existence of a strict local minimum of order two for multiobjective optimization problems with equality and inequality constraints. We suppose that the objective function and the active inequality constraints are only locally Lipschitz. We consider both regular equality constraints and degenerate equality constraints. This article could be considered as a continuation of [E. Constantin, Necessary Conditions for Weak Efficiency for Nonsmooth Degenerate Multiobjective Optimization Problems, J. Global Optim, 75, 111-129, 2019]. We introduce a constraint qualification and a regularity condition, and we show that under each of them, the dual necessary conditions for a weak local minimum of the aforementioned article become of Kuhn-Tucker type.

On variational inequalities using directional convexificators

ABSTRACT In this paper, we give some results which constitute an application of directional convexificators recently introduced by Dempe and Pilecka [Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming. J Global Optim. 2015;61:769–788]. After establishing mean value conditions in terms of directional convexificators, we formulate variational inequalities of Stampacchia and Minty type in terms of directional convexificators and use these variational inequalities as a tool to find out necessary and sufficient conditions for a point to be an optimal solution of an inherent optimization problem. An example illustrating our findings is also given.

Global Solution of Semi-infinite Programs with Existence Constraints

We consider what we term existence-constrained semi-infinite programs. They contain a finite number of (upper-level) variables, a regular objective, and semi-infinite existence constraints. These constraints assert that for all (medial-level) variable values from a set of infinite cardinality, there must exist (lower-level) variable values from a second set that satisfy an inequality. Existence-constrained semi-infinite programs are a generalization of regular semi-infinite programs, possess three rather than two levels, and are found in a number of applications. Building on our previous work on the global solution of semi-infinite programs (Djelassi and Mitsos in J Glob Optim 68(2):227–253, 2017), we propose (for the first time) an algorithm for the global solution of existence-constrained semi-infinite programs absent any convexity or concavity assumptions. The algorithm is guaranteed to terminate with a globally optimal solution with guaranteed feasibility under assumptions that are similar to the ones made in the regular semi-infinite case. In particular, it is assumed that host sets are compact, defining functions are continuous, an appropriate global nonlinear programming subsolver is used, and that there exists a Slater point with respect to the semi-infinite existence constraints. A proof of finite termination is provided. Numerical results are provided for the solution of an adjustable robust design problem from the chemical engineering literature.

An efficient algorithm for large-scale extended trust-region subproblems with non-intersecting linear constraints

In this paper, we study the extended trust-region subproblem in which the trust-region intersects the ball with m linear inequality constraints (m-eTRS). We assume that the linear constraints do not intersect inside the ball. We show that the optimal solution of m-eTRS can be found by solving one TRS, computing the local non-global minimizer of TRS if it exists and solving at most two TRSs with an additional linear equality constraint (1-eqTRS). Both TRS and (1-eqTRS) are polynomially and efficiently solvable, thus the new algorithm significantly improves over the SOCP/SDP relaxation of Burer and Yang [Math Program 149(1-2):253–264, 2015]. on two classes of test problems, the efficiency of the proposed approach is compared with the SOCP/SDP relaxation and branch and bound algorithm of Beck and Pan [J Global Optim 69(2):309–342, 2017].

On the optimality conditions for D.C. vector optimization problems

In this paper, by using the notions of approximate pseudo-dissipativity, strong subdifferential and epsilon-subdifferential, we present sufficient optimality conditions for an epsilon-weak local Pareto minimal solution and an epsilon-local Pareto minimal solution of a vector optimization problem, where its objective function and constraint set are in the form of differences of two vector-valued convex maps, respectively. Then, an application of the given result is obtained for vector fractional programming. Also, some comments are given on two recently published papers presented by Guo and Li [Optimality conditions for vector optimization problems with difference of convex maps. J Optim Theory Appl. 2014;162:821–844] and Gadhi and Metrane [Sufficient optimality condition for vector optimization problems under D.C. J Glob Optim. 2004;28:55–66].

Differential stability of convex discrete optimal control problems with possibly empty solution sets

ABSTRACT As a complement to two recent papers by Toan and Yao (Mordukhovich subgradients of the value function to a parametric discrete optimal control problem. J Global Optim. 2014;58:595–612), and by An and Toan (Differential stability of convex discrete optimal control problems. Acta Math Vietnam. 2018;43:201–217) on subdifferentials of the optimal value function of discrete optimal control problems, this paper studies the differential stability of convex discrete optimal control problems under control constraints, where the solution set may be empty. By using a suitable sum rule for ϵ-subdifferentials and a suitable product rule for ϵ-normal directions, we obtain formulas for computing the ϵ-subdifferential of the optimal value function. Several illustrative examples are also given.

On the relation between the extended supporting hyperplane algorithm and Kelley\u2019s cutting plane algorithm

Recently, Kronqvist et al. (J Global Optim 64(2):249–272, 2016) rediscovered the supporting hyperplane algorithm of Veinott (Oper Res 15(1):147–152, 1967) and demonstrated its computational benefits for solving convex mixed integer nonlinear programs. In this paper we derive the algorithm from a geometric point of view. This enables us to show that the supporting hyperplane algorithm is equivalent to Kelley’s cutting plane algorithm (J Soc Ind Appl Math 8(4):703–712, 1960) applied to a particular reformulation of the problem. As a result, we extend the applicability of the supporting hyperplane algorithm to convex problems represented by a class of general, not necessarily convex nor differentiable, functions.