Localizing vector optimization problems with application to welfare economics

TL;DR

This paper extends local programming techniques to vector optimization problems using Polyak's principle, establishing solution existence and optimality, and applies these results to welfare economics with infinite-dimensional commodity spaces.

Contribution

It introduces a novel application of Polyak's convexity principle to vector optimization and welfare economics, providing new existence and optimality results for localized problems.

Findings

Existence of solutions for localized vector optimization problems.

Conditions for Pareto optimality in infinite-dimensional economies.

Application of local convexity to welfare economics models.

Abstract

In the present paper, the Polyak's principle, concerning convexity of the images of small balls through C1,1 mappings, is employed in the study of vector optimization problems. This leads to extend to such a context achievements of local programming, an approach to nonlinear optimization, due to B.T. Polyak, which consists in exploiting the benefits of the convex local behaviour of certain nonconvex problems. In doing so, solution existence and optimality conditions are established for localizations of vector optimization problems, whose data satisfy proper assumptions. Such results are subsequently applied in the analysis of welfare economics, in the case of an exchange economy model with infinite-dimensional commodity space. In such a setting, the localization of an economy yields existence of Pareto optimal allocations, which, under certain additional assumptions, lead to competitive equilibria.