Fully piecewise linear vector optimization problem

TL;DR

This paper introduces a novel approach to solving fully piecewise linear vector optimization problems by decomposing them into linear subproblems and characterizing their solution sets as unions of polyhedra, extending classical linear optimization results.

Contribution

It provides a new representation for piecewise linear functions and a finite dimensional reduction method for solving fully piecewise linear vector optimization problems, generalizing classical results.

Findings

Solution sets are unions of finitely many generalized polyhedra.

The method applies under mild assumptions and extends classical linear optimization results.

Main results are new even in the linear case.

Abstract

We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed spaces. Based on such a representation, we study a fully piecewise linear vector optimization (PLP) with the objective and constraint functions being piecewise linear. We divide (PLP) into some linear subproblems and structure a finite dimensional reduction method to solve (PLP). Under some mild assumptions, we prove that the Pareto (resp. weak Pareto) solution set of (PLP) is the union of finitely many generalized polyhedra (resp. polyhedra), each of which is contained in a Pareto (resp. weak Pareto) face of some linear subproblem. Our main results are even new in the linear case and further generalize Arrow, Barankin and Blackwell's classical results on linear vector optimization problems in the framework of finite dimensional spaces.