Genericity Results in Linear Conic Programming—A Tour d’Horizon

Abstract

This paper is concerned with so-called generic properties of general linear cone programs. Many results have been obtained on this subject during the last two decades. It has, e.g., been shown in [29] that uniqueness, strict complementarity and nondegeneracy of optimal solutions hold for almost all problem instances. Strong duality holds generically in a stronger sense: it holds for a generic subset of problem instances. In this paper, we survey known results and present new ones. In particular we give an easy proof of the fact that Slater's condition holds generically in linear cone programming. We further discuss the problem of stability of uniqueness, nondegeneracy and strict complementarity. We also comment on the fact that in general, cone programming cannot be treated as a smooth program and that techniques from nonsmooth geometric measure theory are needed.

Verification methods for conic linear programming problems

“…The introduction of redundancies easily happens in this process and this issue affects a wide range of problems occurring in practice. The results in [12,22,23] hold for programming problems with specific conic structures but without consideration of sparsity pattern or typical modeling characteristics. They are therefore not applicable to the limited set of problems that actually occur in practice.…”

“…In [12], Pataki and Tunçel proved this for conic linear programming problems. Recently, Dür, Jargalsaikhan, and Still [22] surveyed different genericity results for this kind of programming problems. They showed that strong duality holds generically in a stronger sense.…”

Recent contributions to linear semi-infinite optimization: an update

“…As a …nal observation in this section, we point out (following [68]) that weak genericity is related with the so-called primal-dual partitions of (see, e.g., [15], [116], [173] and references therein).…”

“…The LSIO theory has also been used in this setting. Indeed, some results in [68], on genericity of strong duality and weak genericity of uniqueness in linear conic optimization (specially those in Subsection 4.1), have been obtained appealing to the LSIO reformulation of P K : However, the known genericity results on LSIO (see [116], [173] and references therein) cannot be directly transferred to linear conic optimization as the class of LSIO reformulations represents a small subset of 0 (see [68,Subsection 4.4]).…”

Recent contributions to linear semi-infinite optimization

“…The LSIO theory has also been used in this setting. Indeed, some results in [59], on genericity of strong duality and weak genericity of uniqueness in linear conic optimization (specially those in Subsection 4.1), have been obtained appealing to the LSIO reformulation of P K : However, the known genericity results on LSIO (see [101], [150] and references therein) cannot be directly transferred to linear conic optimization as the class of LSIO reformulations represents a small subset of 0 (see [59,Subsection 4.4]).…”