Verification methods for conic linear programming problems

Abstract

This paper gives an overview of verification methods for finite dimensional conic linear programming problems. Besides the computation of verified tight enclosures for unique non-degenerate solutions to well-posed conic linear programming instances, we discuss a rigorous treatment of problems with multiple or degenerate solutions. It will be further shown how a priori knowledge about certain boundedness qualifications can be exploited to efficiently compute verified bounds for the optimal objective value. The corresponding approach is applicable even to ill-posed programming problems. Examples from linear and semidefinite programming are used to illustrate the respective approaches and give further explanations. Another topic is the treatment of programming problems whose parameters are subject to uncertainties. Rough but inclusive estimates for the variability of the corresponding programming problems are given, and also a best and worst case analysis is taken into account. At the end of this paper, special consideration is given to typical issues when applying interval arithmetic in the context of conic linear programming. Different ways are shown on how to resolve these issues.