Stochastic Approximation in Convex Multiobjective Optimization

Given a strictly convex multiobjective optimization problem with objective functions

, let us denote by

x_0 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 f 1 , … , f N

with weights

t_1,\ldots,t_N t 1 , … , t N

. The main result of this paper gives an estimation of the averaged error that we make if we approximate

x_0 x 0

with the minimum point of the convex combinations of

n n

functions, chosen among

f_1,\dots,f_N f 1 , … , f N

, with probabilities

t_1,\ldots,t_N t 1 , … , t N

, respectively, and weighted with the same coefficient

1/n 1 / n

. In particular, we prove that the averaged error considered above converges to 0 as

n n

goes to

\infty ∞

, uniformly w.r.t. the weights

t_1,\ldots,t_N t 1 , … , 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 functions

f_1,\dots,f_N f 1 , … , f N

continuously depends on the coefficients of the convex combination.