In this work, we study exact continuous reformulations of nonlinear integer programming problems. To this aim, we preliminarily state conditions to guarantee the equivalence between pairs of general nonlinear problems. Then, we prove that optimal solutions of a nonlinear integer programming problem can be obtained by using various exact penalty formulations of the original problem … Read more
We consider a knapsack problem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in manufacturing and mining, and also appears as a subproblem in decomposition techniques for network design and related problems. We present a new approach for determining facets of the PCKP … Read more
This paper provides a survey of recent progress and software for solving mixed integer nonlinear programs (MINLP) wherein the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have received sustained attention in very years. By exploiting analogies to the case of … Read more
Copositive optimization problems are particular conic programs: extremize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone … Read more
Gomory mixed-integer (GMI) cuts are among the most effective cutting planes for general mixed-integer programs (MIP). They are traditionally generated from an optimal basis of a linear programming (LP) relaxation of an MIP. In this paper we propose a heuristic to generate useful GMI cuts from additional bases of the initial LP relaxation. The cuts … Read more
We consider two-stage quadratic integer programs with stochastic right-hand sides, and present an equivalent reformulation using value functions. We fi rst derive some basic properties of value functions of quadratic integer programs. We then propose a two-phase solution approach. The first phase constructs the value functions of quadratic integer programs in both stages. The second phase … Read more
Computing the 1-width of the incidence matrix of a Steiner Triple System gives rise to small set covering instances that provide a computational challenge for integer programming techniques. One major source of difficulty for instances of this family is their highly symmetric structure, which impairs the performance of most branch-and-bound algorithms. The largest instance in … Read more
In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved … Read more
In this paper we survey recent work on the perspective reformulation approach that generates tight, tractable relaxations for convex mixed integer nonlinear programs (MINLP)s. This preprocessing technique is applicable to cases where the MINLP contains binary indicator variables that force continuous decision variables to take the value 0, or to belong to a convex set. … Read more
In this paper, a variant of the Traveling Salesman Problem with Time Windows is considered, which consists in minimizing the sum of travel durations between a depot and several customer locations. Two mixed integer linear programming formulations are presented for this problem: a classical arc flow model and a sequential assignment model. Several polyhedral results … Read more