Stochastic Approximation in Convex Multiobjective Optimization

Given a strictly convex multiobjective optimization problem with objective functions f1, . . . , fN, let us denote by x0 its solution, obtained as minimum point of the linear scalarized problem, where the objective function is the convex combination of f1, . . . , fN with weights t1, . . . , tN. The main result of this paper gives an estimation of the averaged error that we make if we approximate x0 with the minimum point of the convex combinations of n functions, chosen among f1, . . . , fN, with probabilities t1, . . . , tN, 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 1, uniformly w.r.t. the weights t1, . . . , tN. 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 functions f1, . . . , fN continuously depends on the coefficients of the convex combination.