Applications of reformulations in mathematical programming

Abstract

Mathematical programming is a technique that can be used to solve real-world optimization problems, where one wants to maximize, or minimize, an objective function subject to some constraints on the decision variables. The key features of mathematical programming are the creation of a model for describing the problem (the so called formulation), and the implementation of efficient algorithms to solve it (also called solvers). In this thesis, we focus on the first point. More precisely, we study some problems arising from different domains, and starting from the most natural models for describing them, we propose alternative formulations, which share some properties with the original models but are somehow better (for instance in terms of computational time needed to obtain the solution by the solver). These new models are called reformulations. We follow the classification of reformulations proposed by Liberti in [Reformulations in Mathematical Programming: Definitions and Systematics, RAIRO-OR, 43(1):55-86, 2009]: exact reformulations (also called opt-reformulations), narrowings, relaxations. This thesis is concerned with three mathematical programming applications where the reformulation was crucial to obtain a good solution. The first problem tackled herein is graph clustering by means of modularity maximization. Since this problem is NP-hard, several heuristics are proposed. We focus on a divisive hierarchical algorithm which works by recursively splitting a cluster into two new clusters in an optimal way. This splitting step is performed by solving a convex binary quadratic program. This is reformulated exactly to a more compact form without changing the optimal solutions set (exact reformulation). We also evaluate the impact provided by the reduction of the number of symmetric global optima of the problem, which is also an important topic of the next part of this thesis. The computational times are considerably reduced with respect to the original formulation. The second problem tackled in the thesis is the Packing of Equal Circles in a Square (PECS), where one wants to place non-overlapping equal circles in a unit square in such a way as to maximize the common radius. One of the reasons why the problem is hard to solve is the presence of several symmetric optimal solutions, and consequently a very large Branch-and-Bound tree. Some of the symmetric optima are made infeasible by adjoining some Symmetry Breaking Constraints (SBCs) to the formulation, thereby obtaining a narrowing. Both computational time and size of the Branch-and-Bound tree outperform the ones provided by the original formulation. The third application considered in the thesis is that of computing the convex relaxation for multilinear problems, and to compare the primal formulation and another one obtained using a representation. Although these two relaxations are both already known in the literature, we make a striking observation, i.e., that the dual relaxation leads to a faster and more stable solution process as regards CPU time.