Kernel Density Estimation for Reliable Biobjective Solution of Stochastic Problems

Abstract

Stochastic objective functions can be optimized by finding values in decision space for which the expected output is optimal and the uncertainty is minimal. We investigate the optimization of expensive stochastic black box functions $f: \mathbb {R}^a \times \mathbb {R}^b \rightarrow \mathbb {R}$ with controllable parameter $x \in \mathbb {R}^a$ and b-dimensional random variable C. Our original approach, MIOS-PDE, presented in previous work, requires that the distribution class of C is known. We extend our approach such that it is applicable in situations where the distribution class is unknown. We achieve this with kernel density estimation. With the extended approach, MIOS-KDE, expectation $\mathrm {E}(f(x, C))$ and standard deviation $\mathrm {S}(f(x, C))$ are estimated with a high quality and good approximations of the true Pareto frontiers are obtained. No quality is lost with MIOS-KDE compared to MIOS-PDE. Additionally, MIOS-KDE is robust with respect to outliers, while MIOS-PDE is not.