Necessary and sufficient conditions for linear suboptimality in constrained optimization

This article is devoted to the study of some extremality and optimality notions that are different from conventional concepts of optimal solutions to optimization-related problems. These notions reflect certain amounts of linear subextremality for set systems and linear suboptimality for feasible solutions to multiobjective and scalar optimization problems. In contrast to standard notions of optimality, it is possible to derive necessary and sufficient conditions for linear subextremality and suboptimality in general nonconvex settings, which is done in this article via robust generalized differential constructions of variational analysis in finite-dimensional and infinite-dimensional spaces.      

变分不等式与互补问题、双层规划与平衡约束数学规划问题的若干进展

考虑有限维变分不等式与互补问题、双层规划以及均衡约束的数学规划问题. 在简单介绍这些问题之后,重点介绍近年来这些领域中发展迅速的几个研究方向,包括对称锥互补问题的理论与算法、变分不等式的投影收缩算法、随机变分不等式与随机互补问题的模型与方法、双层规划以及均衡约束数学规划问题的新方法. 最后提出几个进一步研究的方向.     

An interior point potential reduction method for constrained equations

We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algorithm for solving several classes of convex programs. The work of this author was based on research supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739 and the Office of Naval Research under grant N00014-93-1-0228. The work of this author was based on research supported by the National Science Foundation under grant DMI-9496178 and the Office of Naval Research under grants N00014-93-1-0234 and N00014-94-1-0340.     

Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems

In this paper, we introduce a new concept of ϵ-efficiency for vector optimization problems. This extends and unifies various notions of approximate solutions in the literature. Some properties for this new class of approximate solutions are established, and several existence results, as well as nonlinear scalarizations, are obtained by means of the Ekeland’s variational principle. Moreover, under the assumption of generalized subconvex functions, we derive the linear scalarization and the Lagrange multiplier rule for approximate solutions based on the scalarization in Asplund spaces.     

Relative Pareto minimizers for multiobjective problems: existence and optimality conditions

In this paper we introduce and study enhanced notions of relative Pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers for general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient counterparts) that are new in both finite-dimensional and infinite-dimensional settings. Our proofs are based on variational and extremal principles of variational analysis; in particular, on new versions of the Ekeland variational principle and the subdifferential variational principle for set-valued and single-valued mappings in infinite-dimensional spaces.     

Mathematical Programs with Vector Optimization Constraints

In this paper, we introduce mathematical programs with vector optimization constraints. For these problems, we establish two models in both the weak Pareto solution and Pareto solution setting. Some new existence results are obtained under rather weak conditions. We establish also equivalences between mathematical programs with vector optimization constraints and mathematical programs with vector variational inequality constraints.This research was partially supported by a grant from the National Science Council of the ROC. The authors thank the referees for helpful suggestions and comments.     

Methods of variational analysis in multiobjective optimization

This article studies new applications of advanced methods of variational analysis and generalized differentiation to constrained problems of multiobjective/vector optimization. We pay most attention to general notions of optimal solutions for multiobjective problems that are induced by geometric concepts of extremality in variational analysis, while covering various notions of Pareto and other types of optimality/efficiency conventional in multiobjective optimization. Based on the extremal principles in variational analysis and on appropriate tools of generalized differentiation with well-developed calculus rules, we derive necessary optimality conditions for broad classes of constrained multiobjective problems in the framework of infinite-dimensional spaces. Applications of variational techniques in infinite dimensions require certain ‘normal compactness’ properties of sets and set-valued mappings, which play a crucial role in deriving the main results of this article.     

Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities

In this paper, we introduce and study a relaxed extragradient method for finding solutions of a general system of variational inequalities with inverse-strongly monotone mappings in a real Hilbert space. First, this system of variational inequalities is proven to be equivalent to a fixed point problem of nonexpansive mapping. Second, by using the demi-closedness principle for nonexpansive mappings, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient method converges strongly to a solution of this system of variational inequalities. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems. J.-C. Yao’s research was partially supported by a grant from the National Science Council.     

Metric Regularity of Mappings and Generalized Normals to Set Images

The primary goal of this paper is to study some notions of normals to nonconvex sets in finite-dimensional and infinite-dimensional spaces and their images under single-valued and set-valued mappings. The main motivation for our study comes from variational analysis and optimization, where the problems under consideration play a crucial role in many important aspects of generalized differential calculus and applications. Our major results provide precise equality formulas (sometimes just efficient upper estimates) allowing us to compute generalized normals in various senses to direct and inverse images of nonconvex sets under single-valued and set-valued mappings between Banach spaces. The main tools of our analysis revolve around variational principles and the fundamental concept of metric regularity properly modified in this paper.     

Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem

In this paper we develop new applications of variational analysis and generalized differentiation to the following optimization problem and its specifications: given n closed subsets of a Banach space, find such a point for which the sum of its distances to these sets is minimal. This problem can be viewed as an extension of the celebrated Fermat-Torricelli problem: given three points on the plane, find another point that minimizes the sum of its distances to the designated points. The generalized Fermat-Torricelli problem formulated and studied in this paper is of undoubted mathematical interest and is promising for various applications including those frequently arising in location science, optimal networks, etc. Based on advanced tools and recent results of variational analysis and generalized differentiation, we derive necessary as well as necessary and sufficient optimality conditions for the extended version of the Fermat-Torricelli problem under consideration, which allow us to completely solve it in some important settings. Furthermore, we develop and justify a numerical algorithm of the subgradient type to find optimal solutions in convex settings and provide its numerical implementations.     

Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints

In this paper we study set-valued optimization problems with equilibrium constraints (SOPECs) described by parametric generalized equations in the form 0 ∈ G(x) + Q(x), where both G and Q are set-valued mappings between infinite-dimensional spaces. Such models particularly arise from certain optimization-related problems governed by set-valued variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish general results on the existence of optimal solutions under appropriate assumptions of the Palais-Smale type and then derive necessary conditions for optimality in the models under consideration by using advanced tools of variational analysis and generalized differentiation. Dedicated to Jiří V. Outrata on the occasion of his 60th birthday. This research was partly supported by the National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-0451168.     

On system of generalized vector variational inequalities

In this paper, we introduce a new system of generalized vector variational inequalities with variable preference. This extends the model of system of generalized variational inequalities due to Pang and Konnov independently as well as system of vector equilibrium problems due to Ansari, Schaible and Yao. We establish existence of solutions to the new system under weaker conditions that include a new partial diagonally convexity and a weaker notion than continuity. As applications, we derive existence results for both systems of vector variational-like inequalities and vector optimization problems with variable preference.     

Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient for global or local optimality under some MPEC generalized convexity assumptions. Moreover, we propose new constraint qualifications for M-stationary conditions to hold. These new constraint qualifications include piecewise MFCQ, piecewise Slater condition, MPEC weak reverse convex constraint qualification, MPEC Arrow-Hurwicz-Uzawa constraint qualification, MPEC Zangwill constraint qualification, MPEC Kuhn-Tucker constraint qualification, and MPEC Abadie constraint qualification.     

Hybrid inclusion and disclusion systems with applications to equilibria and parametric optimization

In this paper, we first establish the existence theorems of the solution of hybrid inclusion and disclusion systems, from which we study mixed types of systems of generalized quasivariational inclusion and disclusion problems and systems of generalized vector quasiequilibrium problems. Some applications of existence theorems to feasible points for various mathematical programs with variational constraints or equilibrium constraints, system of vector saddle point and system of minimax theorem are also given.     

Iterative methods for variational and complementarity problems

In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.This research was based on work supported by the National Science Foundation under grant ECS-7926320.     

Error bounds for analytic systems and their applications

Using a 1958 result of Lojasiewicz, we establish an error bound for analytic systems consisting of equalities and inequalities defined by real analytic functions. In particular, we show that over any bounded region, the distance from any vectorx in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated atx. For quadratic systems satisfying certain nonnegativity assumptions, we show that this exponent is equal to 1/2. We apply the error bounds to the Karush—Kuhn—Tucker system of a variational inequality, the affine variational inequality, the linear and nonlinear complementarity problem, and the 0–1 integer feasibility problem, and obtain new error bound results for these problems. The latter results extend previous work for polynomial systems and explain why a certain square-root term is needed in an error bound for the (monotone) linear complementarity problem.The research of this author is based on work supported by the Natural Sciences and Engineering Research Council of Canada under grant OPG0090391.The research of this author is based on work supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739 and by the Office of Naval Research under grant 4116687-01.     

Equilibrium Problems under Generalized Convexity and Generalized Monotonicity

Generalized convex functions preserve many valuable properties of mathematical programming problems with convex functions. Generalized monotone maps allow for an extension of existence results for variational inequality problems with monotone maps. Both models are special realizations of an abstract equilibrium problem with numerous applications, especially in equilibrium analysis (e.g., Blum and Oettli, 1994). We survey existence results for equilibrium problems obtained under generalized convexity and generalized monotonicity. We consider both the scalar and the vector case. Finally existence results for a system of vector equilibrium problems under generalized convexity are surveyed which have applications to a system of vector variational inequality problems. Throughout the survey we demonstrate that the results can be obtained without the rigid assumptions of convexity and monotonicity.     

Strong KKT conditions and weak sharp solutions in convex-composite optimization

Using variational analysis techniques, we study convex-composite optimization problems. In connection with such a problem, we introduce several new notions as variances of the classical KKT conditions. These notions are shown to be closely related to the notions of sharp or weak sharp solutions. As applications, we extend some results on metric regularity of inequalities from the convex case to the convex-composite case.     

Equilibrium Problems under Generalized Convexity and Generalized Monotonicity

Generalized convex functions preserve many valuable properties of mathematical programming problems with convex functions. Generalized monotone maps allow for an extension of existence results for variational inequality problems with monotone maps. Both models are special realizations of an abstract equilibrium problem with numerous applications, especially in equilibrium analysis (e.g., Blum and Oettli, 1994). We survey existence results for equilibrium problems obtained under generalized convexity and generalized monotonicity. We consider both the scalar and the vector case. Finally existence results for a system of vector equilibrium problems under generalized convexity are surveyed which have applications to a system of vector variational inequality problems. Throughout the survey we demonstrate that the results can be obtained without the rigid assumptions of convexity and monotonicity.     

Equivalent Lipschitz surrogates for zero-norm and rank optimization problems

This paper proposes a mechanism to produce equivalent Lipschitz surrogates for zero-norm and rank optimization problems by means of the global exact penalty for their equivalent mathematical programs with an equilibrium constraint (MPECs). Specifically, we reformulate these combinatorial problems as equivalent MPECs by the variational characterization of the zero-norm and rank function, show that their penalized problems, yielded by moving the equilibrium constraint into the objective, are the global exact penalization, and obtain the equivalent Lipschitz surrogates by eliminating the dual variable in the global exact penalty. These surrogates, including the popular SCAD function in statistics, are also difference of two convex functions (D.C.) if the function and constraint set involved in zero-norm and rank optimization problems are convex. We illustrate an application by designing a multi-stage convex relaxation approach to the rank plus zero-norm regularized minimization problem.