We give an algorithm for testing the extremality of minimal valid functions for Gomory and Johnson’s infinite group problem, that are piecewise linear (possibly discontinuous) with rational breakpoints. This is the first set of necessary and sufficient conditions that can be tested algorithmically, for deciding extremality in this important class of minimal valid functions. ArticleDownload … Read more
In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MILPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MILPs are a critical element of branch-and-cut. This paper examines the nature of the so-called separation problem, which is … Read more
In case of a special problem class, the simplex method can be implemented as a cutting-plane method that approximates a certain convex polyhedral objective function. In this paper we consider a regularized version of this cutting-plane method, and interpret the resulting procedure as a regularized simplex method. (Regularization is performed in the dual space and … Read more
The feasibility pump is a recent, highly successful heuristic for general mixed integer linear programming problems. We show that the feasibility pump heuristic can be interpreted as a discrete version of the proximal point algorithm. In doing so, we extend and generalize some of the fundamental results in this area to provide new supporting theory. … Read more
This paper introduces novel formulations for optimally responding to epidemics and cyber attacks in networks. In our models, at a given time period, network nodes (e.g., users or computing resources) are associated with probabilities of being infected, and each network edge is associated with some probability of propagating the infection. A decision maker would like … Read more
Power grid vulnerability analysis is often performed through solving a bi-level optimization problem, which, if solved to optimality, yields the most destructive interdiction plan with the worst loss. As one of the most effective operations to mitigate deliberate outages or attacks, transmission line switching recently has been included and modeled by a binary variable in … Read more
In this paper we study the t-branch split cuts introduced by Li and Richard (2008). They presented a family of mixed-integer programs with n integer variables and a single continuous variable and conjectured that the convex hull of integer solutions for any n has unbounded rank with respect to (n-1)-branch split cuts. It was shown … Read more
Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, … Read more
The corner relaxation of a mixed-integer linear program is a central concept in cutting plane theory. In a recent paper Fischetti and Monaci provide an empirical assessment of the strength of the corner and other related relaxations on benchmark problems. In this paper we give a precise characterization of the bounds given by these relaxations … Read more
Gunluk and Pochet [O. Gunluk, Y. Pochet: Mixing mixed integer inequalities. Mathematical Programming 90(2001) 429-457] proposed a procedure to mix mixed integer rounding (MIR) inequalities. The mixed MIR inequalities define the convex hull of the mixing set $\{(y^1,\ldots,y^m,v) \in Z^m \times R_+:\alpha_1 y^i + v \geq \b_i,i=1,\ldots,m\}$ and can also be used to generate valid … Read more