Multiparametric analysis of conic linear optimization based on the lift-and-project procedure

TL;DR

This paper extends the multiparametric analysis of conic linear optimization by introducing a duality framework and invariant region decomposition, providing a unified approach and new insights into optimal partition behavior.

Contribution

It introduces a novel duality concept for mpCLOs and develops an invariant region decomposition, generalizing previous results in multiparametric conic optimization analysis.

Findings

Invariant region decomposition is more general than existing results.

Properties of optimal objective values are characterized as functions of parameters.

The framework unifies and extends previous multiparametric analysis methods.

Abstract

We study how the lift-and-project procedure applies to the multiparametric analysis of conic linear optimization (CLO) problems. We first introduce the concept of a pair of primal and dual conic representable sets and define the set-valued mappings between them. We then explore a novel kind of duality of mpCLOs, which allows us to generalize as well as treat previous results for the mulitparametric analysis in a unified framework. In particular, we discuss the behavior of the optimal partition of a conic representable set. This leads to the invariant region decomposition of a conic representable set that is more general than the known results in the literatures. Finally, we study the properties of the optimal objective values as a function of that parametric vectors. All results are corroborated by examples having correlation.