Reduced quasi-Newton method for simultaneous design and optimization

Adel Hamdi (1, 2) , Andreas Griewank (3)

1 HU Berlin - Humboldt-Universität zu Berlin = Humboldt University of Berlin = Université Humboldt de Berlin
2 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie
3 Institut für Mathematik [Humboldt]

We consider the task of design optimization where the constraint is a state equation that can only be solved by a typically rather slowly converging fixed point solver. This process can be augmented by a corresponding adjoint solver and based on the resulting approximate reduced derivatives also an optimization iteration which actually changes the design. To coordinate the three iterative processes, we use an exact penalty function of doubly augmented Lagrangian type. The main issue here is how to derive a design space preconditioner for the approximated reduced gradient which ensures a consistent reduction of the employed penalty function as well as significant design corrections. Some numerical experiments for an alternating approach where any combination and sequencing of steps are used to improve feasibility and optimality done on a variant of the Bratu problem are presented.

Mots clés: Simultaneous analysis and design; Preconditioning; Augmented Lagrangian; Exact penalty function; Global convergence

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Optimisation et contrôle [math.OC]

Adel Hamdi, Andreas Griewank. Reduced quasi-Newton method for simultaneous design and optimization. Computational Optimization and Applications, 2011, 49 (3), pp.521-548. ⟨10.1007/s10589-009-9306-x⟩. ⟨hal-02426065⟩