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Méthodes primales-duales régularisées pour l'optimisation non linéaire avec contraintes

Abstract : This thesis focuses on the design, analysis, and implementation of efficient and reliable algorithms for solving nonlinearly constrained optimization problems. We present three new strongly primal-dual algorithms to solve such problems. The first feature of these algorithms is that the control of the iterates is done in both primal and dual spaces during the whole minimization process, hence the name “strongly primal-dual”. In particular, the globalization is performed by applying a backtracking line search algorithm based on a primal-dual merit function. The second feature is the introduction of a natural regularization of the linear system solved at each iteration to compute a descent direction. This allows our algorithms to perform well when solving degenerate problems for which the Jacobian of constraints is rank deficient. The third feature is that the penalty parameter is allowed to increase along the inner iterations, while it is usually kept constant. This allows to reduce the number of inner iterations. A detailed theoretical study including the global convergence analysis of both inner and outer iterations, as well as an asymptotic convergence analysis is presented for each algorithm. In particular, we prove that these methods have a high rate of convergence : superlinear or quadratic. These algorithms have been implemented in a new solver for nonlinear optimization which is called SPDOPT. The good practical performances of this solver have been demonstrated by comparing it to the reference codes IPOPT, ALGENCAN and LANCELOT on a large collection of test problems.
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Riadh Omheni. Méthodes primales-duales régularisées pour l'optimisation non linéaire avec contraintes. Optimisation et contrôle [math.OC]. Université de Limoges, 2014. Français. ⟨NNT : 2014LIMO0045⟩. ⟨tel-01136063⟩

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