Convex Quadratic Programming Approach to the Maximum Matching Problem

The problem of determining a maximum matching or whether there exists a perfect matching, is very common in a large variety of applications and as been extensively studied in graph theory. In this paper we start to introduce a characterisation of a family of graphs for which its stability number is determined by convex quadratic programming. The main results connected with the recognition of this family of graphs are also introduced. It follows a necessary and sufficient condition which characterise a graph with a perfect matching and an algorithmic strategy, based on the determination of the stability number of line graphs, by convex quadratic programming, applied to the determination of a perfect matching. A numerical example for the recognition of graphs with a perfect matching is described. Finally, the above algorithmic strategy is extended to the determination of a maximum matching of an arbitrary graph and some related results are presented.

Majorantes para a ordem de subgrafos induzidos k-regulares

Muitos dos problemas de otimização em grafos reduzem-se à determinação de um subconjunto de vértices de cardinalidade máxima que induza um subgrafo k-regular. Uma vez que a determinação da ordem de um subgrafo induzido k-regular de maior ordem é, em geral, um problema NP-difícil, são deduzidos novos majorantes, a determinar em tempo polinomial, que em muitos casos constituam boas aproximações das respetivas soluções ótimas. Introduzem-se majorantes espetrais usando uma abordagem baseada em técnicas de programação convexa e estabelecem-se condições necessárias e suficientes para que sejam atingidos. Adicionalmente, introduzem-se majorantes baseados no espetro das matrizes de adjacência, laplaciana e laplaciana sem sinal. É ainda apresentado um algoritmo não polinomial para a determinação de umsubconjunto de vértices de umgrafo que induz umsubgrafo k-regular de ordem máxima para uma classe particular de grafos. Finalmente, faz-se um estudo computacional comparativo com vários majorantes e apresentam-se algumas conclusões.

Implicit optimality criterion for convex SIP problem with box constrained index set

We consider a convex problem of Semi-Infinite Programming (SIP) with multidimensional index set. In study of this problem we apply the approach suggested in [20] for convex SIP problems with one-dimensional index sets and based on the notions of immobile indices and their immobility orders. For the problem under consideration we formulate optimality conditions that are explicit and have the form of criterion. We compare this criterion with other known optimality conditions for SIP and show its efficiency in the convex case.

Polyhedral study of mixed integer sets arising from inventory problems

“Branch-and-cut” algorithm is one of the most efficient exact approaches to solve mixed integer programs. This algorithm combines the advantages of a pure branch-and-bound approach and cutting planes scheme. Branch-and-cut algorithm computes the linear programming relaxation of the problem at each node of the search tree which is improved by the use of cuts, i.e. by the inclusion of valid inequalities. It should be taken into account that selection of strongest cuts is crucial for their effective use in branch-and-cut algorithm. In this thesis, we focus on the derivation and use of cutting planes to solve general mixed integer problems, and in particular inventory problems combined with other problems such as distribution, supplier selection, vehicle routing, etc. In order to achieve this goal, we first consider substructures (relaxations) of such problems which are obtained by the coherent loss of information. The polyhedral structure of those simpler mixed integer sets is studied to derive strong valid inequalities. Finally those strong inequalities are included in the cutting plane algorithms to solve the general mixed integer problems. We study three mixed integer sets in this dissertation. The first two mixed integer sets arise as a subproblem of the lot-sizing with supplier selection, the network design and the vendor-managed inventory routing problems. These sets are variants of the well-known single node fixed-charge network set where a binary or integer variable is associated with the node. The third set occurs as a subproblem of mixed integer sets where incompatibility between binary variables is considered. We generate families of valid inequalities for those sets, identify classes of facet-defining inequalities, and discuss the separation problems associated with the inequalities. Then cutting plane frameworks are implemented to solve some mixed integer programs. Preliminary computational experiments are presented in this direction.