Algorithm developments for optimization problems with joint reliability constraints

We propose an algorithm for optimization under uncertainty with joint reliability constraints. Most existing research formulates constraints of random variables/parameters in probabilistic forms such that the probability of individual constraint satisfaction must be higher than a reliability target. However, engineering problems generally define reliability as the probability of satisfying constraints 'jointly' rather than individually. Calculating a joint reliability value requires a multiple integration over the feasible domain. This calculation is in most realistic cases impractical by use of exact methods and has become the major challenge in optimization. In this research we propose a filter-based sequential quadratic programming algorithm with joint reliability constraints approximated by their upper bounds. This upper bound can be obtained analytically by considering the current design location, the reliability of satisfying each constraint, and the angles between every constraint pair. To further improve the efficiency of the algorithm, active-set strategies are implemented such that intense reliability calculations only required for constraints that are likely to be active. The rest of the constraints are approximated to the level needed to understand whether constraints might become active in the next iteration. The efficiency of the proposed method enables the applications to general complex engineering problems. Copyright (C) 2010 John Wiley & Sons, Ltd.