w-(H,Ω) CONJUGATE DUALITY THEORY IN MULTIOBJECTIVE NONLINEAR OPTIMIZATION

The duality in multiobjective optimization holds now a major position in the theory of multiobjective programming not only due to its mathematical elegance but also its economic implications. For weak efficiency instead of efficiency, this paper gives the definition and some fundamental properties of the weak supremum and infimum sets. Based on the weak supremum, the concepts, some properties and their relationships of w-(H,Ω) conjugate maps, w-(H,Ω)-subgradients, w- H p Γ (Ω)-regularitions of vector-valued point-to-set maps are provided, and a new duality theory in multiobjective nonlinear optimization------w-(H,Ω) Conjugate Duality Theory is established by means of the w-(H,Ω) conjugate maps. The concepts and their relations to the weak efficient solutions to the primal and dual problems of the w-(H,Ω)-Lagrangian map and weak saddle-point are developed. Finally, several special cases for H and Ω are discussed. Key words: Conjugate duality theory, Multiobjective optimization, Weak efficienc

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

Variational Analysis in Semi-Infinite and Infinite Programming, II: Necessary Optimality Conditions

This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to problems of semi-infinite and infinite programming with feasible solution sets defined by parameterized systems of infinitely many linear inequalities of the type intensively studied in the preceding development [5] from our viewpoint of robust Lipschitzian stability. We present meaningful interpretations and practical examples of such models. The main results establish necessary optimality conditions for a broad class of semi-infinite and infinite programs, where objectives are generally described by nonsmooth and nonconvex functions on Banach spaces and where infinite constraint inequality systems are indexed by arbitrary sets. The results obtained are new in both smooth and nonsmooth settings of semi-infinite and infinite programming

Consideration on Supremum in a Multidimensional Space and Conjugate Duality in Multiobjective Optimization

The first part of this paper is devoted to consideration on the definition of "supremum" in a multi-dimensional Euclidean space. A desirable definition is looked for among several possible alternatives. In the second part conjugate duality in multiobjective optimization is developed. Supremum is defined in the extended multi-dimensional Euclidean space on the basis of consideration in the first part. Some useful concepts such as conjugate maps and subgradients are introduced for vector-valued set-valued maps. Finally a strong duality result for a multiobjective optimization problem is proved under a regularity condition

Variational Analysis in Semi-Infinite and Infinite Programming, II: Necessary Optimality Conditions

This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to problems of semi-infinite and infinite programming with feasible solution sets defined by parameterized systems of infinitely many linear inequalities of the type intensively studied in the preceding development [Cánovas et al., SIAM J. Optim., 20 (2009), pp. 1504–1526] from the viewpoint of robust Lipschitzian stability. The main results establish necessary optimality conditions for broad classes of semi-infinite and infinite programs, where objectives are generally described by nonsmooth and nonconvex functions on Banach spaces and where infinite constraint inequality systems are indexed by arbitrary sets. The results obtained are new in both smooth and nonsmooth settings of semi-infinite and infinite programming. We illustrate our model and results by considering a practically meaningful model of water resource optimization via systems of reservoirs.This research was partially supported by grants MTM2008-06695-C03 (01-02) from MICINN (Spain), ACOMP/2009/047&133, and ACOMP/2010/269 from Generatitat Valenciana (Spain)