Journal of Convex Analysis 31 (2024), No. 3, 761--778
Copyright Heldermann Verlag 2024
Stochastic Approximation in Convex Multiobjective Optimization
Carlo Alberto De Bernardi
Dipartimento di Matematica per le Scienze Economiche, Finanziarie ed Attuariali, Università Cattolica del Sacro Cuore, Milano, Italy
carloalberto.debernardi@unicatt.it
Enrico Miglierina
Dipartimento di Matematica per le Scienze Economiche, Finanziarie ed Attuariali, Università Cattolica del Sacro Cuore, Milano, Italy
enrico.miglierina@unicatt.it
Elena Molho
Dipartimento di Scienze Economiche e Aziendali, Università degli Studi di Pavia, Pavia, Italy
elena.molho@unipv.it
Jacopo Somaglia
Dipartimento di Matematica, Politecnico di Milano, Milano, Italy
jacopo.somaglia@polimi.it
[Abstract-pdf]
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 functions $f_1,\dots,f_N$ continuously depends on the coefficients of the convex combination.
Keywords: Multiobjective optimization, continuity of solution map, convex combinations of convex functions, small diameter property.
MSC: 90C29, 46N10; 90C25.
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