In this paper we present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal” solution of … Read more
Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the right-hand side is random and this random vector has a finite distribution. We give a mixed-integer programming formulation for this special case and study the relaxation … Read more
The classical parameterised integer feasibility problem is as follows. Given a rational matrix $A \in \mathbb{Q}^{m \times n}$ and a rational polyhedron $Q \subseteq \mathbb{R}^m$, decide, whether there exists a point $b \in Q$ such that $A x \leq b$ is integer infeasible. Our main result is a polynomial algorithm to solve a slightly more … Read more
The Generalized Assignment Problem is a well-known NP-hard combinatorial optimization problem which consists of minimizing the assignment costs a set of jobs to a set of machines satisfying capacity constraints. Most of the existing algorithms are based on Branch-and-Price, with lower bounds computed by Dantzig-Wolfe reformulation and column generation. In this paper we propose a … Read more
In this study we consider a multiobjective integer linear stochastic programming problem with individual chance constraints. We assume that there is randomness in the right-hand sides of the constraints only and that the random variables are normally distributed. Some stability notions for such problem are characterized. An auxiliary problem is discussed and an algorithm as … Read more
The United States Postal Service (USPS) delivers more than 200 billion items per year. Transporting these items in a timely and cost-efficient way is a key issue if USPS is to meet its service and financial goals. The Highway Corridor Analytic Program (HCAP) is a tool that aids transportation analysts in identifying cost saving opportunities … Read more
Given a discrete optimization problem $Z(\mb{\tilde{c}})=\max\{\mb{\tilde{c}}’\mb{x}:\mb{x}\in \mathcal{X}\}$, with objective coefficients $\mb{\tilde{c}}$ chosen randomly from a distribution ${\mathcal{\theta}}$, we would like to evaluate the expected value $E_\theta(Z(\mb{\tilde{c}}))$ and the probability $P_{\mathcal{\theta}}(x^*_i(\mb{\tilde{c}})=k)$ where $x^*(\mb{\tilde{c}})$ is an optimal solution to $Z(\mb{\tilde{c}})$. We call this the persistency problem for a discrete optimization problem under uncertain objective, and $P_{\mathcal{\theta}}(x^*_i(\mb{\tilde{c}})=k)$, the … Read more
In this paper we address the following probabilistic version (PSC) of the set covering problem: min { cx | P (Ax>= xi) >= p, x_{j} in {0,1} j in N} where A is a 0-1 matrix, xi is a random 0-1 vector and p in (0,1] is the threshold probability level. In a recent development … Read more
In the context of a variation of the standard UFL (Uncapacitated Facility Location) problem, but with an objective function that is a separable convex quadratic function of the transportation costs, we present some techniques for improving relaxations of MINLP formulations. We use a disaggregation principle and a strategy of developing model-specific valid inequalities (some nonlinear), … Read more
We study the mixed-integer rounding (MIR) closures of polyhedral sets. The MIR closure of a polyhedral set is equal to its split closure and the associated separation problem is NP-hard. We describe a mixed-integer programming (MIP) model with linear constraints and a non-linear objective for separating an arbitrary point from the MIR closure of a … Read more