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

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Listed:
  • Suyun Liu

    (Lehigh University)

  • Luis Nunes Vicente

    (Lehigh University
    Centre for Mathematics of the University of Coimbra (CMUC))

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.

Suggested Citation

  • Suyun Liu & Luis Nunes Vicente, 2023. "Convergence Rates of the Stochastic Alternating Algorithm for Bi-Objective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 165-186, July.
    • Handle: RePEc:spr:joptap:v:198:y:2023:i:1:d:10.1007_s10957-023-02253-w
    DOI: 10.1007/s10957-023-02253-w
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    References listed on IDEAS

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