On an approach to optimization problems with a probabilistic cost and or constraints

We present a new approach to a class of probability constrained optimization problems that arise in the context of optimal engineering design. These problems are characterized by the fact that the probability of failure of one or several components either must be minimized or must not exceed a preassigned threshold. Our approach is interactive: it consists of replacing the original optimal design problem in which either the cost function or a constraint are expressed in terms of a probability of failure, by a constrained minimax problem. Once the minimax problem is solved, the actual probability of failure is computed. Depending on the outcome of this computation, we provide heuristic rules for modifying the minimax problem and repeating this process a couple of times. An important feature of our new approach is that it decouples optimization and probability of failure calculations. This decoupling allows independent selection of methods for the solution of the optimization and the reliability subproblems. We present an example to demonstrate the effectiveness of our approach.