The MVE estimator is an important tool in robust regression and outlier detection in statistics. We develop fast and efficient algorithms for the MVE estimator problem and discuss how they can be implemented efficiently. The novelty of our approach stems from the recent developments in the first-order algorithms for solving the related Minimum Volume Enclosing … Read more
We study a variant of the capacitated vehicle routing problem where the cost over each arc is defined as the product of the arc length and the weight of the vehicle when it traverses that arc. We propose two new mixed integer linear programming formulations for the problem: an arc-load formulation and a set partitioning … Read more
In this paper we present a mathematical programming approach to tighten a Big-$M$ formulation ($P_M$) of a Mixed Integer Problem with Logical Implications ($P$). If $M_0$ is a valid vector (the optimal solutions of $P$ belong to the feasible solutions set of $P_{M_0}$) our procedures find a valid vector $M$ such that $M \leq M_0$. … Read more
We describe a generalization of Benders’ method for solving two-stage stochastic linear optimization problems in which there are both continuous and integer variables in the first and second stages. Benders’ method relies on finding effective lower approximations for the value function of the second-stage problem. In this setting, the value function is a discontinuous, non-convex, … Read more
This paper addresses the value function of a general mixed integer linear optimization problem (MILP). The value function describes the change in optimal objective value as the right-hand side is varied and understanding its structure is central to solving a variety of important classes of optimization problems. We propose a discrete representation of the MILP … Read more
We present two novel applications of symmetries for mixed-integer linear programming. First we propose two variants of a new heuristic to improve the objective value of a feasible solution using symmetries. These heuristics can use either the actual permutations or the orbits of the variables to find better feasible solutions. Then we introduce a new … Read more
We consider a class of multicriteria stochastic optimization problems that features benchmarking constraints based on conditional value-at-risk and second-order stochastic dominance. We develop alternative mixed-integer programming formulations and solution methods for cut generation problems arising in optimization under such multivariate risk constraints. We give the complete linear description of two non-convex substructures appearing in these … Read more
We propose a feasible active set method for convex quadratic programming problems with non-negativity constraints. This method is specifically designed to be embedded into a branch-and-bound algorithm for convex quadratic mixed integer programming problems. The branch-and-bound algorithm generalizes the approach for unconstrained convex quadratic integer programming proposed by Buchheim, Caprara and Lodi to the presence … Read more
This note presents a theoretical analysis of disjunctive constraints featuring unbounded variables. In this framework, classical modeling techniques, including big-M approaches, are not applicable. We introduce a lifted second-order cone formulation of such on/off constraints and discuss related constraint qualification issues. A solution is proposed to avoid solvers’ failure. Citation H. L. Hijazi and L.Liberti … Read more
In this paper, we study bilevel mixed integer programming (MIP) problem and present a novel computing scheme based on reformulations and decomposition strategy. By converting bilevel MIP into a constrained mathematical program, we present its single-level reformulations that are friendly to perform analysis and build insights. Then, we develop a decomposition algorithm based on column-and-constraint … Read more