MIP reformulations of the probabilistic set covering problem

Abstract

In this paper we address the following probabilistic version (PSC) of the set covering problem: min{cx | P(Ax ≥ ξ) ≥ p, x j ∈ {0, 1} N } where A is a 0-1 matrix, ξ is a random 0-1 vector and p ∈ (0, 1] is the threshold probability level. We formulate (PSC) as a mixed integer non-linear program (MINLP) and linearize the resulting (MINLP) to obtain a MIP reformulation. We introduce the concepts of p-inefficiency and polarity cuts. While the former is aimed at reducing the number of constraints in our model, the later is used as a strengthening device to obtain stronger formulations. A hierarchy of relaxations for (PSC) is introduced, and fundamental relationships between the relaxations are established culminating with a MIP reformulation of (PSC) with no additional integer constrained variables. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of ξ are pairwise independent, the distribution function of ξ is a stationary distribution or has the so-called disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a test-bed consisting of almost 10,000 probabilistic instances. This test-bed was created using deterministic instances from the literature and consists of probabilistic variants of the set-covering model and capacitated versions of facility location, warehouse location and k-median models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to fraction of seconds.

Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations

“…The theorem that follows uses polarity to derive strong valid cutting planes for ER. This theorem is motivated by a recent application of the same idea in the context of probabilistic programming [18]. For S ⊆ I, we denote by (< y k > k∈S ) the sub-vector of y having components indexed by S.…”

Branch-and-cut approaches for chance-constrained formulations of reliable network design problems

“…If ξ has a continuous and log-concave distribution, this constraint defines a convex set and hence nonlinear programming techniques can be applied [35,36]. When ξ is a discrete random variable, approaches based on certain nondominated points (known as p-efficient points) of the distribution have been studied [10,18,38]. If ξ further is assumed to have finite support, an approach based on strengthening the corresponding mixed-integer programming formulation using inequalities from [5,21] has been successful [23,28].…”

Models and algorithms for network design problems

“…Although probabilistic constraints have been widely studied for many years, see Henrion (2004);Prékopa (2003); and the references therein, papers on problems with integer variables are not very numerous. Among them, problems featuring joint probabilistic constraints with random right hand side have been studied by Ruszczynski (2002b,a, 2005) who propose exact and heuristic branch-and-bound algorithms, Dentcheva et al (2002) who study formulations and bounding procedures, Lejeune and Ruszczynski (2007) who develop a column-generation based algorithm for a supply chain management problem, and Saxena et al (2009) who introduce the concepts of p-ine ciency and provide extensive computational results for the probabilistic set-covering problem studied by Beraldi and Ruszczynski (2002a). All these works handle probabilistic constraints through the concept of p-e cient points introduced by Prékopa (1990), apart from Saxena et al (2009) who uses p-ine cient points instead.…”