Motivated by stochastic 0-1 integer programming problems with an expected utility objective, we study the mixed-integer nonlinear set: $P = \cset{(w,x)\in \reals \times \set{0,1}^N}{w \leq f(a’x + d), b’x \leq B}$ where $N$ is a positive integer, $f:\reals \mapsto \reals$ is a concave function, $a, b \in \reals^N$ are nonnegative vectors, $d$ is a real … Read more
In this paper we derive strong linear inequalities for sets of the form {(x, q) ∈ R^d × R : q ≥ Q(x), x ∈ R^d − int(P ) }, where Q(x) : R^d → R is a quadratic function, P ⊂ R^d and “int” denotes interior. Of particular but not exclusive interest is the … Read more
This paper introduces a new cutting plane method for two-stage stochastic mixed-integer programming (SMIP) called Fenchel decomposition (FD). FD uses a class of valid inequalities termed, FD cuts, which are derived based on Fenchel cutting planes from integer programming. First, we derive FD cuts based on both the first and second-stage variables, and devise an … Read more
We extend work on generating relaxations of simple sets with guaranteed bounds. Citation Unpublished. Columbia University, July 2008. Article Download View Tightening simple mixed-integer sets with guaranteed bounds
Recently, Andersen et al.(2007), Borozan and Cornuejols (2007) and Cornuejols and Margot(2007) characterized extreme inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts (Balas (1971)) derived using maximal lattice-free convex sets. In order to use these inequalities to obtain … Read more
We consider the vehicle routing problem where one can choose among vehicles with different costs and capacities to serve the trips. We develop six different formulations: the first four based on Miller-Tucker-Zemlin constraints and the last two based on flows. We compare the linear programming bounds of these formulations. We derive valid inequalities and lift … Read more
We consider the problem of installing a two-level telecommunication network. Terminal nodes communicate with each other through hubs. Hubs can be installed on terminal nodes and they are interconnected by a complete network. Each terminal is connected to a hub node by direct links. The aim is to minimize the cost of installing hubs and … Read more
In this study, a lifting procedure is applied to some existing formulations of the Diameter Constrained Minimum Spanning Tree Problem. This problem typically models network design applications where all vertices must communicate with each other at minimum cost, while meeting or surpassing a given quality requirement. An alternative formulation is also proposed for instances of … Read more
We consider knapsack constraints involving one general integer and many binary variables. We introduce the concept of a cover for such a constraint and we construct a new family of valid inequalities based on this concept. We generalize this idea to extended covers, and we propose a specialized lifting procedure for cover inequalities. Finally, we … Read more
In this paper we discuss the generation of strong valid inequalities for (mixed) integer knapsack sets based on lifting of valid inequalities for basic knapsack sets with two integer variables (and one continuous variable). The description of the basic polyhedra can be made in polynomial time. We use superadditive valid functions in order to obtain … Read more