Metric regularity of the feasible set mapping in semi-infinite optimization

For parametric systems of (finitely many) equations and (infinitely many) inequalities the well-known concept of metric regularity is shown to be equivalent to the so-called extended Mangasarian-Fromovitz constraint qualification. By this, a corresponding result obtained by Robinson for finite optimization problems my be transferred to semi-infinite optimization. For the proof a local epigraph representation of the constraint set is mainly used.     

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.     

An optimization result for set-valued mappings and a stability property in vector problems with constraints

First, we prove an existence result relative to minimal points of set-valued mappings. Then, conditions about the upper and lower semicontinuity of constraint sets defined through set-valued mappings are given. Finally, a stability result relative to vector problems with abstract constraints is proved.The author thanks the referee for helpful comments on the first version of this paper.     

Scalarizing vector optimization problems

A scalarization of vector optimization problems is proposed, where optimality is defined through convex cones. By varying the parameters of the scalar problem, it is possible to find all vector optima from the scalar ones. Moreover, it is shown that, under mild assumptions, the dependence is differentiable for smooth objective maps defined over reflexive Banach spaces. A sufficiency condition of optimality for a general mathematical programming problem is also given in the Appendix.     

Necessary optimality conditions for a class of nonsmooth vector optimization

The Kuhn-Tucker type necessary conditions of weak efficiency are given for the problem of minimizing a vector function whose each component is the sum of a differentiable function and a convex function, subject to a set of differentiable nonlinear inequalities on a convex subset C of ℝ ⁿ , under the conditions similar to the Abadie constraint qualification, or the Kuhn-Tucker constraint qualification, or the Arrow-Hurwicz-Uzawa constraint qualification. Supported by the National Natural Science Foundation of China (No. 70671064, No. 60673177), the Province Natural Science Foundation of Zhejiang (No.Y7080184) and the Education Department Foundation of Zhejiang Province (No. 20070306).     

Existence theorems in vector optimization

In this paper, existence theorems for minimal, weakly minimal, and properly minimal elements of a subset of a partially-ordered, real linear space are presented.This paper was written when the author was a visitor at the Department of Mathematics, North Carolina State University, Raleigh, North Carolina.     

Sectionwise connected sets in vector optimization

We introduce the notion of sectionwise connected set as a new tool to investigate nonconvex vector optimization. Indeed, the image of a K-convex set through a K-quasiconnected vector function is proved to be sectionwise connected. Some properties of the minimal frontiers of sectionwise connected sets are studied.     

Complexity of vector optimization problems on graphs

The paper is dedicated to the computation complexity of multi-objective optimization problems on graphs. The classes of multi-objective problems with polynomial complexity or being polynomially reduced to be NP-hard are marked out. The unsolvability of a series of combinatorial multi-objective problems has been set up by means of linear convolution algorithm. The sufficient conditions under which these algorithms are statistically efficient have also been obtained.     

Criteria for efficiency in vector optimization

We consider unconstrained finite dimensional multi-criteria optimization problems, where the objective functions are continuously differentiable. Motivated by previous work of Brosowski and da Silva (1994), we suggest a number of tests (TEST 1–4) to detect, whether a certain point is a locally (weakly) efficient solution for the underlying vector optimization problem or not. Our aim is to show: the points, at which none of the TESTs 1–4 can be applied, form a nowhere dense set in the state space. TESTs 1 and 2 are exactly those proposed by Brosowski and da Silva. TEST 3 deals with a local constant behavior of at least one of the objective functions. TEST 4 includes some conditions on the gradients of objective functions satisfied locally around the point of interest. It is formulated as a Conjecture. It is proven under additional assumptions on the objective functions, such as linear independence of the gradients, convexity or directional monotonicity. This work was partially supported by grant 55681 of the CONACyT.     

Qualification conditions for multivalued functions in Banach spaces with applications to nonsmooth vector optimization problems

In this paper we introduce qualification conditions for multivalued functions in Banach spaces involving the A-approximate subdifferential, and we show that these conditions guarantee metric regularity of multivalued functions. The results are then applied for deriving Lagrange multipliers of Fritz—John type and Kuhn—Tucker type for infinite non-smooth vector optimization problems.     

Some equivalent problems in set optimization

Given a set-valued optimization problem (P), there is more than one way of defining the solutions associated with it. Depending on the decision maker’s preference, we consider the vector criterion or the set criterion. Both criteria of solution are considered together to solve problem (P) by reducing the feasible set.     

Well-posedness and convexity in vector optimization

We study a notion of well-posedness in vector optimization through the behaviour of minimizing sequences of sets, defined in terms of Hausdorff set-convergence. We show that the notion of strict efficiency is related to the notion of well-posedness. Using the obtained results we identify a class of well-posed vector optimization problems: the convex problems with compact efficient frontiers.     

Necessary optimality conditions in nondifferentiable vector optimization

The purpose of the present paper is to give necessary optimality conditions for weak Pareto minimun, peints of nondifferentiable vector optimization problems vcing generalized definitions of the upper and lower Dini-Hadamard derivatives. We give two different approaches for such definitions, a global one and a componentwise one     

Robust optimization revisited via robust vector Farkas lemmas

This paper provides characterizations of the weakly minimal elements of vector optimization problems and the global minima of scalar optimization problems posed on locally convex spaces whose objective functions are deterministic while the uncertain constraints are treated under the robust (or risk-averse) approach, i.e. requiring the feasibility of the decisions to be taken for any possible scenario. To get these optimality conditions we provide Farkas-type results characterizing the inclusion of the robust feasible set into the solution set of some system involving the objective function and possibly uncertain parameters. In the particular case of scalar convex optimization problems, we characterize the optimality conditions in terms of the convexity and closedness of an associated set regarding a suitable point.     

Continuous perturbations of infinite optimization problems

In this paper, continuity properties of the extremal value function and the solution function are studied for general optimization problems with perturbations in the objective function and the constraints. A classical stability condition is extended and compared with constraint qualification conditions.     

Super-efficiency of vector optimization in Banach spaces

Using variational analysis, in terms of the Clarke normal cone, we consider super-efficiency of vector optimization in Banach spaces. We establish some characterizations for super-efficiency. In particular, dropping the assumption that the ordering cone has a bounded base, we extend a result in Borwein and Zhuang [J.M. Borwein, D. Zhuang, Super-efficiency in vector optimization, Trans. Amer. Math. Soc. 338 (1993) 105-122] to the nonconvex setting.     

Approximate proximal methods in vector optimization

This paper studies the vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C ⊂ Y with a nonempty interior. The proximal method in vector optimization is extended to develop an approximate proximal method for this problem by virtue of the approximate proximal point method for finding a root of a maximal monotone operator. In this approximate proximal method, the subproblems consist of finding weakly efficient points for suitable regularizations of the original mapping. We present both an absolute and a relative version, in which the subproblems are solved only approximately. Weak convergence of the generated sequence to a weak efficient point is established. In addition, we also discuss an extension to Bregman-function-based proximal algorithms for finding weakly efficient points for mappings.     

Sensitivity analysis in vector optimization

For a vector optimization problem that depends on a parameter vector, the sensitivity analysis of perturbation, proper perturbation, and weak perturbation maps is dealth with. Each of the perturbation maps is defined as a set-valued map which associates to each parameter value the set of all minimal, properly minimal, and weakly minimal points of the perturbed feasible set in the objective space with respect to a fixed ordering cone. Using contingent cones in a finite-dimensional Euclidean space, we investigate the relationship between the contingent derivatives of the three types of perturbation maps and three types of minimal point sets for the contingent derivative of the feasible-set map in the objective space. These results provide quantitative informations on the behavior of the perturbation maps.The authors would like to thank the referees for their valuable comments and suggestions.     

Existence of solutions for vector optimization

In this paper, we prove the existence of a weak minimum for constrained vector optimization problem by making use of vector variational-like inequality and preinvex functions.     

Conjugate duality in vector optimization and some applications to the vector variational inequality

The aim of this paper is to extend the so-called perturbation approach in order to deal with conjugate duality for constrained vector optimization problems. To this end we use two conjugacy notions introduced in the past in the literature in the framework of set-valued optimization. As a particular case we consider a vector variational inequality which we rewrite in the form of a vector optimization problem. The conjugate vector duals introduced in the first part allow us to introduce new gap functions for the vector variational inequality. The properties in the definition of the gap functions are verified by using the weak and strong duality theorems.