Relaxations of the bilinear term, $x_1x_2=x_3$, play a central role in constructing relaxations of factorable functions. This is because they can be used directly to relax products of functions with known relaxations. In this paper, we provide a compact, closed-form description of the convex hull of this and other more general bivariate monomial terms (which … Read more
This paper concerns a method of selecting the best subset of explanatory variables for a linear regression model. Employing Mallows’ C_p as a goodness-of-fit measure, we formulate the subset selection problem as a mixed integer quadratic programming problem. Computational results demonstrate that our method provides the best subset of variables in a few seconds when … Read more
The Reformulation-Linearization Technique (RLT), due to Sherali and Adams, can be used to construct hierarchies of linear programming relaxations of mixed 0-1 polynomial programs. As one moves up the hierarchy, the relaxations grow stronger, but the number of variables increases exponentially. We present a procedure that generates cutting planes at any given level of the … Read more
We study the problem of integrated staffing and scheduling under demand uncertainty. The problem is formulated as a two-stage stochastic integer program with mixed-integer recourse. The here-and-now decision is to find initial staffing levels and schedules, well ahead in time. The wait-and-see decision is to adjust these schedules at a time epoch closer to the … Read more
We present a fast branch-and-bound algorithm for solving convex quadratic integer programs with few linear constraints. In each node, we solve the dual problem of the continuous relaxation using an infeasible active set method proposed by Kunisch and Rendl to get a lower bound; this active set algorithm is well suited for reoptimization. Our algorithm … Read more
We study the approximability of the classical quadratic knapsack problem (QKP) on special graph classes. In this case the quadratic terms of the objective function are not given for each pair of knapsack items. Instead an edge weighted graph G = (V,E) whose vertices represent the knapsack items induces a quadratic profit p_ij for the … Read more
We present a coarse-grain (outer-loop) parallel implementation of RLT1/2/3 (Level 1, 2, and 3 Reformulation and Linearization Technique—in that order) bound calculations for the QAP within a branch-and-bound procedure. For a search tree node of size S, each RLT3 and RLT2 bound calculation iteration is parallelized S ways, with each of S processors performing O(S5) … Read more
In this paper we introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective function. The underestimators are the novelty of the approach because the standard approaches … Read more
We present a control problem for an electrical vehicle. Its motor can be operated in two discrete modes, leading either to acceleration and energy consumption, or to a recharging of the battery. Mathematically, this leads to a mixed-integer optimal control problem (MIOCP) with a discrete feasible set for the controls taking into account the electrical … Read more
Mathematical Programming is Turing complete, and can be used as a general-purpose declarative language. We present a new constructive proof of this fact, and showcase its usefulness by discussing an application to finding the hardest input of any given program running on a Minsky Register Machine. We also discuss an application of Mathematical Programming to … Read more