Stochastic Approximation in convex multiobjective optimization

Abstract

Given a strictly convex multiobjective optimization problem with objective functions $f_1,\dots,f_N$, let us denote by $x_0$ its solution, obtained as minimum point of the linear scalarized problem, where the objective function is the convex combination of $f_1,\dots,f_N$ with weights $t_1,\ldots,t_N$. The main result of this paper gives an estimation of the averaged error that we make if we approximate $x_0$ with the minimum point of the convex combinations of $n$ functions, chosen among $f_1,\dots,f_N$, with probabilities $t_1,\ldots,t_N$, respectively, and weighted with the same coefficient $1/n$. In particular, we prove that the averaged error considered above converges to 0 as $n$ goes to $\infty$, uniformly w.r.t. the weights $t_1,\ldots,t_N$. The key tool in the proof of our stochastic approximation theorem is a geometrical property, called by us small diameter property, ensuring that the minimum point of a convex combination of the function $f_1,\dots,f_N$ continuously depends on the coefficients of the convex combination.