Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions.     

An interior proximal method in vector optimization

This paper studies the vector optimization problem of finding weakly efficient points for maps from Rⁿ to R^m, with respect to the partial order induced by a closed, convex, and pointed cone C⊂R^m, with nonempty interior. We develop for this problem an extension of the proximal point method for scalar-valued convex optimization problem with a modified convergence sensing condition that allows us to construct an interior proximal method for solving VOP on nonpolyhedral set.     

Newton-like methods for efficient solutions in vector optimization

In this work we study the Newton-like methods for finding efficient solutions of the vector optimization problem for a map from a finite dimensional Hilbert space X to a Banach space Y, with respect to the partial order induced by a closed, convex and pointed cone C with a nonempty interior. We present both exact and inexact versions, in which the subproblems are solved approximately, within a tolerance. Furthermore, we prove that under reasonable hypotheses, the sequence generated by our method converges to an efficient solution of this problem.     

A generic approach to approximate efficiency and applications to vector optimization with set-valued maps

In this paper we focus on approximate minimal points of a set in Hausdorff locally convex spaces. Our aim is to develop a general framework from which it is possible to deduce important properties of these points by applying simple results. For this purpose we introduce a new concept of ε-efficient point based on set-valued mappings and we obtain existence results and properties on the behavior of these approximate efficient points when ε is fixed and by considering that ε tends to zero. Finally, the obtained results are applied to vector optimization problems with set-valued mappings.     

Approximate methods for convex minimization problems with series–parallel structure

Consider a problem of minimizing a separable, strictly convex, monotone and differentiable function on a convex polyhedron generated by a system of m linear inequalities. The problem has a series–parallel structure, with the variables divided serially into n disjoint subsets, whose elements are considered in parallel. This special structure is exploited in two algorithms proposed here for the approximate solution of the problem. The first algorithm solves at most min{m, ν − n + 1} subproblems; each subproblem has exactly one equality constraint and at most n variables. The second algorithm solves a dynamically generated sequence of subproblems; each subproblem has at most ν − n + 1 equality constraints, where ν is the total number of variables. To solve these subproblems both algorithms use the authors’ Projected Newton Bracketing method for linearly constrained convex minimization, in conjunction with the steepest descent method. We report the results of numerical experiments for both algorithms.     

Some remarks on the Minty vector variational principle

In scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Kluwer Academic, Dordrecht, 1997, pp. 93-99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193-201]. In these papers, in the particular case of a differentiable objective function f taking values in Rm and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions.     

Kuratowski convergence of the efficient sets in the parametric linear vector semi-infinite optimization

In the space of whole linear vector semi-infinite optimization problems we consider the mappings putting into correspondence to each problem the set of efficient and weakly efficient points, respectively. We endow the image space with Kuratowski convergence and by means of the lower and upper semi-continuity of these mappings we prove generic well-posedness of the vector optimization problems. The connection between the continuity and some properties of the efficient sets is also discussed.     

Coefficient bounds for biholomorphic mappings which have a parametric representation

The main purpose of this paper is to prove a collection of new fixed point theorems for so-called weakly F-contractive mappings. By analogy, we introduce also a class of strongly F-expansive mappings and we prove fixed point theorems for such mappings. We provide a few examples, which illustrate these results and, as an application, we prove an existence and uniqueness theorem for the generalized Fredholm integral equation of the second kind. Finally, in Appendix A, we apply the Mönch fixed point theorem to prove two results on the existence of approximate fixed points of some continuous mappings.     

Geoffrion type characterization of higher-order properly efficient points in vector optimization

We consider the constrained vector optimization problem minCf(x), x∈A, where X and Y are normed spaces, A⊂X₀⊂X are given sets, C⊂Y, C≠Y, is a closed convex cone, and is a given function. We recall the notion of a properly efficient point (p-minimizer) for the considered problem and in terms of the so-called oriented distance we define also the notion of a properly efficient point of order n (p-minimizers of order n). We show that the p-minimizers of higher order generalize the usual notion of a properly efficient point. The main result is the characterization of the p-minimizers of higher order in terms of “trade-offs.” In such a way we generalize the result of A.M. Geoffrion [A.M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl. 22 (3) (1968) 618-630] in two directions, namely for properly efficient points of higher order in infinite dimensional spaces, and for arbitrary closed convex ordering cones.     

Inertial forward–backward methods for solving vector optimization problems

Abstract

We propose two forward–backward proximal point type algorithms with inertial/memory effects for determining weakly efficient solutions to a vector optimization problem consisting in vector-minimizing with respect to a given closed convex pointed cone the sum of a proper cone-convex vector function with a cone-convex differentiable one, both mapping from a Hilbert space to a Banach one. Inexact versions of the algorithms, more suitable for implementation, are provided as well, while as a byproduct one can also derive a forward–backward method for solving the mentioned problem. Numerical experiments with the proposed methods are carried out in the context of solving a portfolio optimization problem.     

Generalized Proximal Method for Efficient Solutions in Vector Optimization

This article is devoted to developing the generalized proximal algorithm of finding efficient solutions to the vector optimization problem for a mapping from a uniformly convex and uniformly smooth Banach space to a real Banach space with respect to the partial order induced by a pointed closed convex cone. In contrast to most published literature on this subject, our algorithm does not depend on the nonemptiness of ordering cone of the space under consideration and deals with finding efficient solutions of the vector optimization problem in question. We prove that under some suitable conditions the sequence generated by our method weakly converges to an efficient solution of this problem.     

Tikhonov-type regularization method for efficient solutions in vector optimization

The paper is devoted to developing the Tikhonov-type regularization algorithm of finding efficient solutions to the vector optimization problem for a mapping between finite dimensional Hilbert spaces with respect to the partial order induced by a pointed closed convex cone. We prove that under some suitable conditions either the sequence generated by our method converges to an efficient solution or all of its cluster points belong to the set of all efficient solutions of this problem.     

Derivatives of the Efficient Point Multifunction in Parametric Vector Optimization Problems

In this paper, we deal with the sensitivity analysis in vector optimization. More specifically, formulae for inner and outer evaluating the S-derivative of the efficient point multifunction in parametric vector optimization problems are established. These estimating formulae are presented via the set of efficient/weakly efficient points of the S-derivative of the original multifunction, a composite multifunction of the objective function and the constraint mapping. The elaboration of the formulae in vector optimization problems, having multifunction constraints and semiinfinite constraints, is also undertaken. Furthermore, examples are provided for analyzing and illustrating the obtained results.     

A proximal point algorithm for the monotone second-order cone complementarity problem

This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by Pan and Chen (Optimization 59:1173–1197, 2010), which confirm the theoretical results and the effectiveness of the algorithm.     

Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems

We consider an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of N nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal methods. We derive a necessary and sufficient condition for weak convergence of the sequences generated by the proposed scheme.     

Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality.We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting.     

Exponential type vector variational-like inequalities and vector optimization problems with exponential type invexities

A new class of exponential form of vector variational-like inequality problems is introduced, and then the equivalence among (weakly) efficient solutions, vector critical points of vector optimization problem and the solutions of vector variational-like inequalities under the framework of (p,r)-invexity is established. To the best our knowledge, the presented results are new and highly application oriented to other results based on generalized invexities to the context of optimization problems in the literature.     

On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces

In vector optimization, topological properties of the set of efficient and weakly efficient points are of interest. In this paper, we study the connectedness of the setE ^w of all weakly efficient points of a subsetZ of a locally convex spaceX with respect to a continuous mappingp:X Y,Y locally convex and partially ordered by a closed, convex cone with nonempty interior. Under the general assumptions thatZ is convex and closed and thatp is a pointwise quasiconvex mapping (i.e., a generalized quasiconvex concept), the setE ^w is connected, if the lower level sets ofp are compact. Furthermore, we show some connectedness results on the efficient points and the efficient and weakly efficient outcomes. The considerations of this paper extend the previous results of Refs. 1–3. Moreover, some examples in vector approximation are given.The author is grateful to Dr. D. T. Luc and to a referee for pointing out an error in an earlier version of this paper.     

Convergence Analysis of a Class of Penalty Methods for Vector Optimization Problems with Cone Constraints

In this paper, we consider convergence properties of a class of penalization methods for a general vector optimization problem with cone constraints in infinite dimensional spaces. Under certain assumptions, we show that any efficient point of the cone constrained vector optimization problem can be approached by a sequence of efficient points of the penalty problems. We also show, on the other hand, that any limit point of a sequence of approximate efficient solutions to the penalty problems is a weekly efficient solution of the original cone constrained vector optimization problem. Finally, when the constrained space is of finite dimension, we show that any limit point of a sequence of stationary points of the penalty problems is a KKT stationary point of the original cone constrained vector optimization problem if Mangasarian–Fromovitz constraint qualification holds at the limit point.This work is supported by the Postdoctoral Fellowship of Hong Kong Polytechnic University.