Homotopy methods for constraint relaxation in unilevel reliability based design optimization

Reliability based design optimization is a methodology for finding optimized designs that are characterized with a low probability of failure. The main ob jective in reliability based design optimization is to minimize a merit function while satisfying the reliability constraints. The reliability constraints are constraints on the probability of failure corre- sponding to each of the failure modes of the system or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a relia- bility analysis. During the last few years, a variety of different techniques have been devel- oped for reliability based design optimization. Traditionally, these have been formulated as a double-loop (nested) optimization problem. The upper level optimization loop gen- erally involves optimizing a merit function sub ject to reliability constraints and the lower level optimization loop(s) compute the probabilities of failure corresponding to the failure mode(s) that govern the system failure. This formulation is, by nature, computationally intensive. A new efficient unilevel formulation for reliability based design optimization was developed by the authors in earlier studies. In this formulation, the lower level optimiza- tion (evaluation of reliability constraints in the double loop formulation) was replaced by its corresponding first order Karush-Kuhn-Tucker (KKT) necessary optimality conditions at the upper level optimization. It was shown that the unilevel formulation is computation- ally equivalent to solving the original nested optimization if the lower level optimization is solved by numerically satisfying the KKT conditions (which is typically the case), and the two formulations are mathematically equivalent under constraint qualification and general- ized convexity assumptions. In the unilevel formulation, the KKT conditions of the inner optimization for each probabilistic constraint evaluation are imposed at the system level as equality constraints. Most commercial optimizers are usually numerically unreliable when applied to problems accompanied by many equality constraints. In this investigation an optimization framework for reliability based design using the unilevel formulation is de- veloped. Homotopy methods are used for constraint relaxation and to obtain a relaxed feasible design. A series of optimization problems are solved as the relaxed optimization problem is transformed via a homotopy to the original problem. A heuristic scheme is employed in this paper to update the homotopy parameter. The proposed algorithm is illustrated with example problems.