Convergence Rates of the Stochastic Alternating Algorithm for Bi-Objective Optimization

Suyun Liu () and Luis Nunes Vicente ()
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Suyun Liu: Lehigh University
Luis Nunes Vicente: Lehigh University

Journal of Optimization Theory and Applications, 2023, vol. 198, issue 1, No 6, 165-186

Abstract: Abstract Stochastic alternating algorithms for bi-objective optimization are considered when optimizing two conflicting functions for which optimization steps have to be applied separately for each function. Such algorithms consist of applying a certain number of steps of gradient or subgradient descent on each single objective at each iteration. In this paper, we show that stochastic alternating algorithms achieve a sublinear convergence rate of $${\mathcal {O}}(1/T)$$ O ( 1 / T ) , under strong convexity, for the determination of a minimizer of a weighted-sum of the two functions, parameterized by the number of steps applied on each of them. An extension to the convex case is presented for which the rate weakens to $${\mathcal {O}}(1/\sqrt{T})$$ O ( 1 / T ) . These rates are valid also in the non-smooth case. Importantly, by varying the proportion of steps applied to each function, one can determine an approximation to the Pareto front.

Keywords: Multi-objective optimization; Pareto front; Stochastic optimization; Alternating optimization (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-023-02253-w

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