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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.

In “Asymmetric Multidepot Vehicle Routing Problems: Valid Inequalities and a Branch-and-Cut Algorithm,” Uit het Broek, Schrotenboer, Jargalsaikhan, Roodbergen, and Coelho present a generic branch-and-cut framework to solve routing problems with multiple depots on directed graphs. They present new valid inequalities that eliminate subtours, enforce tours to be linked to the same depot, and enforce bounds on the number of customers in a vehicle tour. This is embedded in a branch-and-cut scheme that also contains generalized and adapted versions of valid inequalities that are well known for related routing problems. The authors show that the new inequalities tighten root node relaxations considerably. In combination with a simple but effective upper-bound procedure, only requiring a MIP solver and a smart reduction of the problem size, the authors show that the overall framework solves instances of considerably larger size to optimality than have been reported in the literature.

Abstract. Checking copositivity of a matrix is a co-NP-complete problem. This paper studies copositive matrices with certain spectral properties. It shows that an indefinite matrix with exactly one positive eigenvalue is copositive if and only if the matrix is nonnegative. Moreover, it shows that finding out if a matrix with exactly one negative eigenvalue is strictly copositive or not can be formulated as a combination of two convex quadratic programming problems which can be solved efficiently.