Descent Methods for Design Optimization under Uncertainty

This paper is about optimization under uncertainty, when the uncertain parameters are modeled through random variables. Contrary to traditional robust approaches, which deal with a deterministic problem through a worst-case scenario formulation, the stochastic algorithms presented introduce the distribution of the random variables modeling the uncertainty. For single-objective problems such methods are currently classical, based on the Robbins-Monro algorithm. When several objectives are involved, the optimization problem becomes much more difficult and the few available methods in the literature are based on a genetic approach coupled with Monte-Carlo approaches, which are numerically very expensive. We present a new algorithm for solving the expectation formulation of stochastic smooth or non-smooth multi-objective optimization problems. The proposed method is an extension of the classical stochastic gradient algorithm to multi-objective optimization, using the properties of a common descent vector. The mean square and the almost-certain convergence of the algorithm are proven. The algorithm efficiency is illustrated and assessed on an academic example.