Détection de la non-réalisabilité et stratégies de régularisation en optimisation non linéaire

Abstract : This thesis is devoted to the study of numerical algorithms for nonlinear optimization. On the one hand, we propose new strategies for the rapid infeasibility detection. On the other hand, we analyze the local behavior of primal-dual algorithms for the solution of singular problems. In the first part, we present a modification of an augmented Lagrangian algorithm for equality constrained optimization. The quadratic convergence of the new algorithm in the infeasible case is theoretically and numerically demonstrated. The second part is dedicated to extending the previous result to the solution of general nonlinear optimization problems with equality and inequality constraints. We propose a modification of a mixed logarithmic barrier-augmented Lagrangian algorithm. The theoretical convergence results and the numerical experiments show the advantage of the new algorithm for the infeasibility detection. In the third part, we study the local behavior of a primal-dual interior point algorithm for bound constrained optimization. The local analysis is done without the standard assumption of the second-order sufficient optimality conditions. These conditions are replaced by a weaker assumption based on a local error bound condition. We propose a regularization technique of the Jacobian matrix of the optimality system. We then demonstrate some boundedness properties of the inverse of these regularized matrices, which allow us to prove the superlinear convergence of our algorithm. The last part is devoted to the local convergence analysis of the primal-dual algorithm used in the first two parts of this thesis. In practice, it has been observed that this algorithm converges rapidly even in the case where the constraints do not satisfy the Mangasarian-Fromovitz constraint qualification. We demonstrate the superlinear and quadratic convergence of this algorithm without any assumption of constraint qualification.
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Ngoc Nguyen Tran. Détection de la non-réalisabilité et stratégies de régularisation en optimisation non linéaire. Optimization and Control [math.OC]. Université de Limoges, 2018. English. ⟨NNT : 2018LIMO0059⟩. ⟨tel-01930327⟩

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