Abstract

Given a strictly convex multiobjective optimization problem with objective functions f1,…,fNf_1,\dots,f_N, let us denote by x0x_0 its solution, obtained as minimum point of the linear scalarized problem, where the objective function is the convex combination of f1,…,fNf_1,\dots,f_N with weights t1,…,tNt_1,\ldots,t_N. The main result of this paper gives an estimation of the averaged error that we make if we approximate x0x_0 with the minimum point of the convex combinations of nn functions, chosen among f1,…,fNf_1,\dots,f_N, with probabilities t1,…,tNt_1,\ldots,t_N, respectively, and weighted with the same coefficient 1/n1/n. In particular, we prove that the averaged error considered above converges to 0 as nn goes to ∞\infty, uniformly w.r.t. the weights t1,…,tNt_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 f1,…,fNf_1,\dots,f_N continuously depends on the coefficients of the convex combination.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Multiobjective optimization
  • continuity of solution map
  • convex combinations of convex functions
  • small diameter property

Mathematics Subject Classification

90C29, 46N10; 90C25

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Suggested citation

C. A. De Bernardi, E. Miglierina, E. Molho, J. Somaglia. “Stochastic Approximation in Convex Multiobjective Optimization.” Journal of Convex Analysis 31 (2024), No. 3, 761–778. https://doi.org/10.68381/jca31037

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.