Proyecto FONDECYT 1100667
Marzo 2010 - Febrero 2012
Fabián Flores-bazán [P]:
Unifying General Vector Optimization Problems and Optimality Conditions
This project deals with a theoretical treatment of vector optimization problems, mostly with weakly efficient (or efficient) solutions on one side; and with preference relations which are reflexive and transitive on the other. It starts by introducing and analysing a general vector optimization problem and one of its approximate which encompass those related to efficiency, weak efficiency, and many others notions of efficiency including those related to properness (a special type of efficiency), in a unified framework. By a carefully analysis of a well known nonlinear scalarizing function, new characterizations of solution (complete scalarizations) and new optimality conditions for both problems (the original and the approximate) via subdifferentials in the standar convex and nonconvex cases, are established. In the latter situation the Mordukhovich subdifferential is used. New equivalent formulations of Gordan-type alternative theorems are also presented via quasi-relative interior. When bicriteria problems are considered, complete characterizations of the weakly as well as the (Benson) proper efficient solution set through their linear scalarizations, are established. Applications to characterize the strong duality property for cone constrained nonconvex optimization problems involving cones having possibly empty interior, are also treated. In particular, a new complete characterization of this property for a problem with a single inequality constraint is provided: showing that strong duality still holds without the standard Slater condition. In addition, a general Brezis-Browder principle involving reflexive and transitive preference relations is established. Thus, various versions of Ekeland-variational principle are derived, and applications to vector optimization problems with set-valued mappings are also considered. Moreover, it is also provided a condition under which the (weak Pareto) weakly efficient and the (Pareto) efficient solution sets coincide. This is very important for applications, besides production theory, also in optimal portfolio and bargaining theory. Closedness or free-disposability is not imposed.