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Theses

Global optimization of polynomial programs with mixed-integer variables

Abstract : In this thesis, we are interested in the study of polynomial programs, that is optimization problems for which the objective function and/or the constraints are expressed by multivariate polynomials. These problems have many practical applications and are currently actively studied. Different methods can be used to find either a global or a heuristic solution, using for instance, positive semi-definite relaxations as in the "Moment/Sum of squares" method. But these problems remain very difficult and only small instances are addressed. In the quadratic case, an effective exact solution approach was initially proposed in the QCR method. It is based on a quadratic convex reformulation, which is optimal in terms of continuous relaxation bound.One of the motivations of this thesis is to generalize this approach to the case of polynomial programs. In most of this manuscript, we study optimization problems with binary variables. We propose two families of convex reformulations for these problems: "direct" reformulations and quadratic ones.For direct reformulations, we first focus on linearizations. We introduce the concept of q-linearization, that is a linearization using q additional variables, and we compare the bounds obtained by continuous relaxation for different values of q. Then, we apply convex reformulation to the polynomial problem, by adding additional terms to the objective function, but without adding additional variables or constraints.The second family of convex reformulations aims at extending quadratic convex reformulation to the polynomial case. We propose several new alternative reformulations that we compare to existing methods on instances of the literature. In particular we present the algorithm PQCR to solve unconstrained binary polynomial problems. The PQCR method is able to solve several unsolved instances. In addition to numerical experiments, we also propose a theoretical study to compare the different quadratic reformulations of the literature and then apply a convex reformulation to them.Finally, we consider more general problems and we propose a method to compute convex relaxations for continuous problems.
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Arnaud Lazare. Global optimization of polynomial programs with mixed-integer variables. Optimization and Control [math.OC]. Université Paris-Saclay, 2019. English. ⟨NNT : 2019SACLY011⟩. ⟨tel-02462537v2⟩

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