First order solutions in conic programming

Mirjam Dür,Georg Still,Bolor Jargalsaikhan

doi:10.1007/s00186-015-0513-1

Abstract

We study the order of maximizers in linear conic programming (CP) as well as stability issues related to this. We do this by taking a semi-infinite view on conic programs: a linear conic problem can be formulated as a special instance of a linear semi-infinite program (SIP), for which characterizations of the stability of first order maximizers are well-known. However, conic problems are highly special SIPs, and therefore these general SIP-results are not valid for CP. We discuss the differences between CP and general SIP concerning the structure and results for stability of first order maximizers, and we present necessary and sufficient conditions for the stability of first order maximizers in CP.

Highlights

We consider linear conic problems (CP) of the form n max cT x s.t
We present approriate conditions for CP and provide necessary and sufficient conditions for the stability of first order maximizers in CP
We further present some examples of first order maximizers for Semidefinite programming (SDP) and Copositive programming (COP)

Summary

Introduction

We consider linear conic problems (CP) of the form n max cT x s.t. X := B − xi Ai ∈ K,. Given K and n, m ∈ N, the set of instances of conic problems (P) and (D) is parametrized by CP =. In what follows we are interested in characterizations of optimizers of order 1 and their stability. In semi-infinite optimization (SIP), characterizations of first order solutions as well as their stability properties are well-studied, see e.g., Fischer (1991), Goberna and Lopez (1998), Nürnberger (1985), Hettich and Kortanek (1993), Helbig and Todorov (1998). The aim of this paper is to express and interpret the characterizations of first order maximizers for SIP in the context of conic programming, and to analyse the stability behaviour of first order maximizers. 3, we discuss different characterizations of first order maximizers in terms of conic programming.

Preliminaries

Stability of first order maximizers