Reliable Biobjective Solution of Stochastic Problems Using Metamodels

Abstract

For many real world optimization problems, the objective function is stochastic. When optimizing a stochastic function f, one has to deal with the problem of varying outputs f(x, C) for the same input x due to the effects of a random variable C. One possibility for optimizing f is considering the expectation and standard deviation of f(x, C) and choosing x such that the expected value of f(x, C) is optimal, e.g. minimal and the standard deviation of f(x, C) is minimal. This turns the optimization of f into a biobjective optimization problem. We investigate the optimization of expensive stochastic black box functions f(x, C) with $x \in \mathbb {R}$ and C being a one dimensional random variable. Because f is an expensive function, we want to evaluate it seldom. Therefore, we use a surrogate model $\hat{f}$ of f and numerical integration to estimate the expectation $\mathrm {E}(f(x, C))$ and the standard deviation $\mathrm {S}(f(x, C))$. We perform a simulation study to analyze how well our approach works and compare it to a classic method. Our approach enables us to estimate $\mathrm {E}(f(x, C))$ and $\mathrm {S}(f(x, C))$ for each feasible x-value with a comparably high quality and yields a good approximation of the true Pareto set at the cost of requiring that C is observable.