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Hdr Année : 2019

Some contributions to mathematical programming and combinatorial optimization

Résumé

Consider the optimization (i.e. maximization or minimization) of a real valued function f defined on a subset of the n-dimensional real vector space, subject to some set of constraints on the variables. The present document gathers and summarizes part of the work I could carry out on such optimization problems when the objective function is either linear or quadratic and when the constraints set on the variables are linear and/or integrality constraints. In the first part, methods are presented to deal with general families of mathematical programs: (mixed-integer) linear programs and unconstrained binary quadratic programs, namely. Chapter 1 is concerned with different types of cutting-plane methods for (mixed-integer) linear programs. In Chapter 2 we present a combinatorial algorithm to solve a particular family of unconstrained binary quadratic programs in polynomial time. General unconstrained quadratic binary programs are then considered in Chapter 3, for which we show how information related to the spectrum of the matrix defining the objective may be used to derive bounds on the optimal objective value. In the second part, we present our contributions w.r.t. several classical combinatorial optimization problems. They namely contain studies on descriptions of polytopes related to the maximum cut problem when the sizes of the shores are fixed (Chapter 4). Given an edge weighted graph, this problem consists in identifying a node subset S of prescribed size, such that the sum of the weights of the edges having exactly one endpoint in S is maximized. Another very classical problem in combinatorial optimization is the one which consists in identifying a minimum cardinality dominating set. Recall that given a graph, a dominating set is a node subset S such that each node not in S has at least one neighbor in S. Polyhedral investigations have been carried out w.r.t. the weighted version of the problem and for different generalizations of the domination concept (Chapter 5). A graph partitioning problem called the k-separator problem is then dealt with from different perspectives (complexity, approximations and polyhedral descriptions) in Chapter 6. Basically it consists in identifying a node subset in a graph whose removal induces a graph having connected components of bounded size. Finally, in Chapter 7, we present several features of an original orientation problem in graphs, including the identification of polynomial time solvable cases, several formulations and connections with the maximum cut problem.
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tel-04261549 , version 1 (27-10-2023)

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José Neto. Some contributions to mathematical programming and combinatorial optimization. Computer Science [cs]. Université Evry-Val d'Essonne/Université Paris Saclay, 2019. ⟨tel-04261549⟩
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