Genericity Results in Linear Conic Programming - A Tour d'Horizon

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.