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.     

Necessary conditions for super minimizers in constrained multiobjective optimization

This paper concerns the study of the so-called super minimizers related to the concept of super efficiency in constrained problems of multiobjective optimization, where cost mappings are generally set-valued. We derive necessary conditions for super minimizers on the base of advanced tools of variational analysis and generalized differentiation that are new in both finite-dimensional and infinite-dimensional settings for problems with single-valued and set-valued objectives.      

Necessary Conditions in Nonsmooth Minimization via Lower and Upper Subgradients

The paper concerns first-order necessary optimality conditions for problems of minimizing nonsmooth functions under various constraints in infinite-dimensional spaces. Based on advanced tools of variational analysis and generalized differential calculus, we derive general results of two independent types called lower subdifferential and upper subdifferential optimality conditions. The former ones involve basic/limiting subgradients of cost functions, while the latter conditions are expressed via Fréchet/regular upper subgradients in fairly general settings. All the upper subdifferential and major lower subdifferential optimality conditions obtained in the paper are new even in finite dimensions. We give applications of general optimality conditions to mathematical programs with equilibrium constraints deriving new results for this important class of intrinsically nonsmooth optimization problems.     

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.     

Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs

The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical Mangasarian–Fromovitz and Farkas–Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for exact calculations of the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation. In the further development we derive first-order necessary optimality conditions for infinite and semi-infinite programs, which are new in both finite-dimensional and infinite-dimensional settings.     

Necessary Conditions in Multiobjective Optimization with Equilibrium Constraints

We study multiobjective optimization problems with equilibrium constraints (MOPECs) described by parametric generalized equations in the form

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.      

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.     

Variational Analysis of Marginal Functions with Applications to Bilevel Programming

This paper pursues a twofold goal. First goal is to derive new results on generalized differentiation in variational analysis focusing mainly on a broad class of intrinsically nondifferentiable marginal/value functions. Then the results established in this direction are applied to deriving necessary optimality conditions for the optimistic version of bilevel programs, which occupy a remarkable place in optimization theory and its various applications. We obtain new sets of optimality conditions in both smooth and nonsmooth settings of finite-dimensional and infinite-dimensional spaces.     

Characterizations of linear suboptimality for mathematical programs with equilibrium constraints

The paper is devoted to the study of a new notion of linear suboptimality in constrained mathematical programming. This concept is different from conventional notions of solutions to optimization-related problems, while seems to be natural and significant from the viewpoint of modern variational analysis and applications. In contrast to standard notions, it admits complete characterizations via appropriate constructions of generalized differentiation in nonconvex settings. In this paper we mainly focus on various classes of mathematical programs with equilibrium constraints (MPECs), whose principal role has been well recognized in optimization theory and its applications. Based on robust generalized differential calculus, we derive new results giving pointwise necessary and sufficient conditions for linear suboptimality in general MPECs and its important specifications involving variational and quasivariational inequalities, implicit complementarity problems, etc. Research was partially supported by the National Science Foundation under grant DMS-0304989 and by the Australian Research Council under grant DP-0451168.     

Nonsmooth Multiobjective Problems and Generalized Vector Variational Inequalities Using Quasi-Efficiency

In this paper, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Here, a generalized Stampacchia vector variational inequality is formulated as a tool to characterize quasi- or weak quasi-efficient points. By using two new classes of generalized convexity functions, under suitable constraint qualifications, the equivalence between Kuhn–Tucker vector critical points, solutions to the multiobjective problem and solutions to the generalized Stampacchia vector variational inequality in both weak and strong forms will be proved.     

Calculus of sequential normal compactness in variational analysis

In this paper we study some properties of sets, set-valued mappings, and extended-real-valued functions unified under the name of “sequential normal compactness.” These properties automatically hold in finite-dimensional spaces, while they play a major role in infinite-dimensional variational analysis. In particular, they are essential for calculus rules involving generalized differential constructions, for stability and metric regularity results and their broad applications, for necessary optimality conditions in constrained optimization and optimal control, etc. This paper contains principal results ensuring the preservation of sequential normal compactness properties under various operations over sets, set-valued mappings, and functions.     

优化和均衡的等价性

通过向量优化问题, 向量变分不等式问题以及向量变分原理来分析优化问题及均衡问题的一致性.从而显然, 可以用统一的观点来处理数值优化、向量优化以及博弈论等问题.进而为非线性分析提供了一个新的发展空间.     

Conjugate dual problems in constrained set-valued optimization and applications

In this paper, two conjugate dual problems are proposed by considering the different perturbations to a set-valued vector optimization problem with explicit constraints. The weak duality, inclusion relations between the image sets of dual problems, strong duality and stability criteria are investigated. Some applications to so-called variational principles for a generalized vector equilibrium problem are shown.     

多目标优化正则条件的一个注记

给出带不等式约束的非光滑多目标优化问题正则条件的一个例子.通过该例,指出最近由Burachik和Rizvi利用线性化锥提出的可微多目标优化问题的正则条件不能利用Clarke导数推广到非光滑情形.     

Affine Variational Inequalities on Normed Spaces

This paper studies infinite-dimensional affine variational inequalities on normed spaces. It is shown that infinite-dimensional quadratic programming problems and infinite-dimensional linear fractional vector optimization problems can be studied by using affine variational inequalities. We present two basic facts about infinite-dimensional affine variational inequalities: the Lagrange multiplier rule and the solution set decomposition.     

Second-order variational analysis and characterizations of tilt-stable optimal solutions in infinite-dimensional spaces

The paper is devoted to developing second-order tools of variational analysis and their applications to characterizing tilt-stable local minimizers of constrained optimization problems infinite-dimensional spaces with many results new also in finite-dimensional settings. The importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization. Based on second-order generalized differentiation, we obtain qualitative and quantitative characterizations of tilt stability in general frameworks of constrained optimization and establish its relationships with strong metric regularity of subgradient mappings and uniform second-order growth. The results obtained are applied to deriving new necessary and sufficient conditions for tilt-stable minimizers in problems of nonlinear programming with twice continuously differentiable data in Hilbert spaces.     

Equilibrium existence theorems for multi-leader-follower generalized multiobjective games in FC-spaces

In this paper, we introduce and study a class of multi-leader-follower generalized multiobjective games in FC-spaces where the number of leaders and followers may be finite or infinite and the objective functions of leaders and followers get their values in infinite-dimensional spaces. By using a Pareto equilibrium existence theorem of generalized constrained multiobjective games in FC-spaces due to author, some equilibrium existence theorems for the multi-leader-follower generalized multiobjective games are established in noncompact FC-spaces. These results improve and generalize some corresponding results in recent literatures.     

Generalized Homotopy Approach to Multiobjective Optimization

This paper proposes a new generalized homotopy algorithm for the solution of multiobjective optimization problems with equality constraints. We consider the set of Pareto candidates as a differentiable manifold and construct a local chart which is fitted to the local geometry of this Pareto manifold. New Pareto candidates are generated by evaluating the local chart numerically. The method is capable of solving multiobjective optimization problems with an arbitrary number k of objectives, makes it possible to generate all types of Pareto optimal solutions, and is able to produce a homogeneous discretization of the Pareto set. The paper gives a necessary and sufficient condition for the set of Pareto candidates to form a (k-1)-dimensional differentiable manifold, provides the numerical details of the proposed algorithm, and applies the method to two multiobjective sample problems.     

Enhanced metric regularity and Lipschitzian properties of variational systems

This paper mainly concerns the study of a large class of variational systems governed by parametric generalized equations, which encompass variational and hemivariational inequalities, complementarity problems, first-order optimality conditions, and other optimization-related models important for optimization theory and applications. An efficient approach to these issues has been developed in our preceding work (Aragón Artacho and Mordukhovich in Nonlinear Anal 72:1149–1170, 2010) establishing qualitative and quantitative relationships between conventional metric regularity/subregularity and Lipschitzian/calmness properties in the framework of parametric generalized equations in arbitrary Banach spaces. This paper provides, on one hand, significant extensions of the major results in op.cit. to partial metric regularity and to the new hemiregularity property. On the other hand, we establish enhanced relationships between certain strong counterparts of metric regularity/hemiregularity and single-valued Lipschitzian localizations. The results obtained are new in both finite-dimensional and infinite-dimensional settings.