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Guaranteed Deterministic Global Optimization using Constraint Programming through Algebraic, Functional and Piecewise Differential Constraints

Hugo Joudrier 1
1 G-SCOP_ROSP [2016-2019] - Recherche Opérationnelle pour les Systèmes de Production [2016-2019]
G-SCOP - Laboratoire des sciences pour la conception, l'optimisation et la production
Abstract : In this thesis a set of tools based on guaranteed methods are presented in order to solve multi-physics dynamic problems. These systems can be applied in various domains such that engineering design process, model of chemical reactions, simulation of biological systems or even to predict athletic performances.The resolution of these optimization problems is made of two stages. The first one consists in defining a mathematical model by setting up the equations for the problem. The model is made of a set of variables, a set of algebraic and functional constraints and cost functions. The latter are used in the second stage in order to extract the optimal solutions from the model depending on several criteria (volume, weight, etc).Algebraic constraints are used to describe the static properties of the system (quantity, size, density, etc). They are non-linear, non-convex and sometimes discontinuous. Functional constraints are used to manipulate dynamic quantities. These constraints can be quite simple such as monotony or periodicity or they can be more complex such as simple or piecewise differential constraints. Differential equations are used to describe physico-chemical properties (magnetic, thermal, etc) and other features evolving with the component use. Several levels of approximation exist for each of these two stages. These approximations give some relevant results but they do not guarantee the feasibility nor the optimality of the solutions.After presenting a set of guaranteed methods in order to perform the guaranteed integration of ordinary differential equations, a peculiar type of hybrid system that can be modeled with piecewise ordinary differential equation is considered. A new method that computes guaranteed integration of these piecewise ordinary differential equations is developed through an extension of the initial algorithm based on several proofs and theorems. In a second step these algorithms are gathered within a contractor programming module that have been implemented. It is used to solve algebraic and functional constraint satisfaction problems with guaranteed methods. Finally, the considered optimization problems are solved with a modular deterministic global optimization algorithm that uses the previous modules.
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Hugo Joudrier. Guaranteed Deterministic Global Optimization using Constraint Programming through Algebraic, Functional and Piecewise Differential Constraints. Methods and statistics. Université Grenoble Alpes, 2018. English. ⟨NNT : 2018GREAI010⟩. ⟨tel-01872573⟩

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