A Minty variational principle for set optimization

Extremal problems are studied involving an objective function with values in (order) complete lattices of sets generated by so-called set relations. Contrary to the popular paradigm in vector optimization, the solution concept for such problems, introduced by F. Heyde and A. Löhne, comprises the attainment of the infimum as well as a minimality property. The main result is a Minty type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions. The variational inequality is based on a new Dini directional derivative for set-valued functions which is defined in terms of a “lattice difference quotient.” A residual operation in a lattice of sets replaces the inverse addition in linear spaces. Relationships to families of scalar problems are pointed out and used for proofs. The appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as Diewert's theorem to improper functions.     

广义梯度计算公式的推广

本文用分析中的几何方法推广了D.E.Ward的广义梯度计算公式,给出一个基本计算定理.     

Local-global properties of bifunctions

Representations of composite systems, such as bilinear programming, models of consumer/producer behavior, and sensitivity problems involve bifunctions (functions of two vector arguments). Such bifunctions are typically convex, pseudoconvex, or quasiconvex in each of their arguments, but not jointly convex, pseudoconvex, or quasiconvex. These functions do not in general possess the strong local-global property, namely, that every stationary point is a global minimum. In this paper, we define conditions that ensure that a bifunction possesses only a global minimum. In exploring this question, we use P-convexity and pseudo P-convexity, which are classes of bifunctions that generalize quasiconvexity and pseudoconvexity.     

Optimality Conditions in Differentiable Multiobjective Programming

In this paper, three sufficient conditions are given, one of which modifies the previous result given by Singh (Ref. 1) under the assumption of convexity of the functions involved at the Pareto-optimal solution. A counterexample has been furnished which shows that the convexity assumption cannot be extended to include the quasiconvexity case. The second theorem on sufficiency requires the strict pseudoconvexity of the functions involved.     

Note on generalized convex functions

In this note, an important class of generalized convex functions, called invex functions, is defined under a general framework, and some properties of the functions in this class are derived. It is also shown that a function is (generalized) pseudoconvex if and only if it is quasiconvex and invex.     

A note on quasi-equilibrium problems

The purpose of this paper is to prove the existence of solutions of quasi-equilibrium problems without any generalized monotonicity assumption. Additionally, we give some applications.     

A modified Fritz John optimality criterion

A modified Fritz John optimality criterion is presented for a certain class of nonlinear programming problems.This research was partially supported by AFOSR Grant No. AF-AFOSR-78-3518.     

Directional derivatives in nonsmooth optimization

In this note, we consider two notions of second-order directional derivatives and discuss their use in the characterization of minimal points for nonsmooth functions.This research was supported by NSF Grant No. ECS-8214081, by the Fund for Promotion of Research at the Technion, and by Deutsche Forschungsgemeinschaft.     

Generalized monotonicity and generalized convexity

Generalized monotonocity of bifunctions or multifunctions is a rather new concept in optimization and nonsmooth analysis. It is shown in the present paper how quasiconvexity, pseudoconvexity, and strict pseudoconvexity of lower semicontinuous functions can be characterized via the quasimonotonicity, pseudomonotonicity, and strict pseudomonotonicity of different types of generalized derivatives, including the Dini, Dini-Hadamard, Clarke, and Rockafellar derivatives as well.This research was supported by the National Science Foundation of Hungary, Grant No. OTKA 1313/1991.     

A note on NTU convexity

For cooperative games with transferable utility, convexity has turned out to be an important and widely applicable concept. Convexity can be defined in a number of ways, each having its own specific attractions. Basically, these definitions fall into two categories, namely those based on a supermodular interpretation and those based on a marginalistic interpretation. For games with nontransferable utility, however, the literature mainly focuses on two kinds of convexity, ordinal and cardinal convexity, which both extend the supermodular interpretation. In this paper, we analyse three types of convexity for NTU games that generalise the marginalistic interpretation of convexity. Received: December 2000     

A survey of recent developments in multiobjective optimization

Multiobjective Optimization (MO) has many applications in such fields as the Internet, finance, biomedicine, management science, game theory and engineering. However, solving MO problems is not an easy task. Searching for all Pareto optimal solutions is expensive and a time consuming process because there are usually exponentially large (or infinite) Pareto optimal solutions. Even for simple problems determining whether a point belongs to the Pareto set is -hard. In this paper, we discuss recent developments in MO. These include optimality conditions, applications, global optimization techniques, the new concept of epsilon Pareto optimal solution, and heuristics.     

A note on functions whose local minima are global

In this note, we introduce a new class of generalized convex functions and show that a real functionf which is continuous on a compact convex subsetM of ⁿ and whose set of global minimizers onM is arcwise-connected has the property that every local minimum is global if, and only if,f belongs to that class of functions.     

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.     

Geodesic convexity in nonlinear optimization

The properties of geodesic convex functions defined on a connected RiemannianC ² k-manifold are investigated in order to extend some results of convex optimization problems to nonlinear ones, whose feasible region is given by equalities and by inequalities and is a subset of a nonlinear space.This research was supported in part by the Hungarian National Research Foundation, Grant No. OTKA-1044.     

Necessary Optimality Conditions in Terms of Convexificators in Lipschitz Optimization

This study is devoted to constraint qualifications and Kuhn-Tucker type necessary optimality conditions for nonsmooth optimization problems involving locally Lipschitz functions. The main tool of the study is the concept of convexificators. First, the case of a minimization problem in the presence of an arbitrary set constraint is considered by using the contingent cone and the adjacent cone to the constraint set. Then, in the case of a minimization problem with inequality constraints, Abadie type constraint qualifications and several other qualifications are proposed; Kuhn-Tucker type necessary optimality conditions are derived under the qualifications.Communicated by S. SchaibleThe authors thank the referees for bringing to their attention some papers closely related to this study and for helpful comments and constructive suggestions that have greatly improved the original version of the paper. Further, they are indebted to Professors H. W. Sun and F. Y. Lu, who suggested an example for this paper. The first author thanks S. Schaible for encouragement during this research.     

G-Semidifferentiability in Euclidean Spaces

To unify the increasing number of generalized derivatives in the recent literature, a general concept of differentiability, embracing several classical cases, has been proposed in Ref. 1. Proceeding in this vein, the present note is devoted to deepening the study of the properties of this general approach and to point out, as much as possible, the connections with the Dini–Hadamard derivatives. The results obtained in this sense lay the groundwork for a future comparison among several different optimality conditions for constrained extremum problems.     

A characterization of weakly efficient points

In this paper, we study a characterization of weakly efficient solutions of Multiobjective Optimization Problems (MOPs). We find that, under some quasiconvex conditions of the objective functions in a convex set of constraints, weakly efficient solutions of an MOP can be characterized as an optimal solution to a scalar constraint problem, in which one of the objectives is optimized and the remaining objectives are set up as constraints. This characterization is much less restrictive than those found in the literature up to now.Corresponding author.     

Some results on pseudolinear quadratic fractional functions

In the paper, the affinity of quadratic fractional functions and the gradient of pseudolinear quadratic fractional functions are characterized. This research was supported in part by the Hungarian Scientific Research Fund, Grant No. OTKA-T043276 and OTKA-K60480.