Homotopy methods for constraint relaxation in unilevel reliability based design optimization

Abstract

In reliability based design optimization, a methodology for finding optimized designs characterized with a low probability of failure the main objective is to minimize a merit function while satisfying the reliability constraints. Traditionally, these have been formulated as a double-loop (nested) optimization problem, which is computationally intensive. A new efficient unilevel formulation for reliability based design optimization was developed by the authors in earlier studies, where the lower-level optimization was replaced by its corresponding first-order Karush–Kuhn–Tucker (KKT) necessary optimality conditions at the upper-level optimization and imposed as equality constraints. But as most commercial optimizers are usually numerically unreliable when applied to problems accompanied by many equality constraints, an optimization framework for reliability based design using the unilevel formulation is developed here. Homotopy methods are used for constraint relaxation and to obtain a relaxed feasible design and heuristic scheme is employed to update the homotopy parameter.

Keywords:

• reliability based robust design
• necessary optimality condition
• convergent unilevel formulation
• homotopy method
• performance measure approach

Acknowledgements

This research was supported in part by the Center for Applied Mathematics at the University of Notre Dame; NSF Grants DMI01-1497, DMI-0422719, DMI-0355391; Department of Energy Grant DE-FG02-06ER25720; and ONR Grant N00014-02-1-0786.