Multipoint linear approximations for stochastic chance constrained optimization problems with integer design variables

In multidisciplinary analysis and optimization response surface approximations are frequently applied. An important reason is that response surface techniques provide a convenient representation of data from one discipline to other disciplines [1]. In each discipline usually one or more computationally expensive computer simulation models are involved. The response surface approximations are used to alleviate the computational burden of the optimization and circumvent discontinuities or noisy responses that are encountered. They also enable to deal in a practical way with discrete design variables: the discete optimization problem can be solved on the approximate optimization level which avoids a combinatorial increase of the number of numerical analyses (see e.g. [2]). The computational burden becomes even larger if the system behaves stochastic instead of deterministic. One may think of analysis for reliability or discrete-event simulation, where stochastic distributions are part of the modeling. In that case the stochasticity has to be accounted for during the optimization. Response surface techniques have their origin in physical experimentation and are therefore especially suitable to build deterministic approximate optimization subproblems from stochastic data. The optimum of the deterministic approximate subproblem then has to be evaluated in a statistical sense with respect to feasibility and change of objective function value. If no further information is available on the actual distributions, a statistical evaluation usually requires several replications of the numerical analysis of the proposed approximate optimum design. This paper addresses integer optimization problems with stochastic objective function and constraints. Starting point is a sequential approximate optimization approach, where response-surface techniques are employed to build multipoint linear approximations of objective function and constraints in search subregions of the design space. The research question is whether such an approach is able to solve a stochastic integer optimization problem in a practical valuable way within a manageable number of computer simulations. The focus of this paper lies on optimization problems with so-called chance constraints. This type of constraints typically arise for structures, such as aircraft, where conditions on, for example, reliability are included.