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Incremental Constraint Projection Methods for Monotone Stochastic Variational Inequalities

Author

Listed:
  • Alfredo N. Iusem

    (Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, RJ, Brazil)

  • Alejandro Jofré

    (Centro de Modelamiento Matemático (CMM and DIM), Universidad de Chile, Santiago de Chile, Chile)

  • Philip Thompson

    (Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, RJ, Brazil)

Abstract

We consider stochastic variational inequalities (VIs) with monotone operators where the feasible set is an intersection of a large number of convex sets. We propose a stochastic approximation method with incremental constraint projections, meaning that a projection method is taken after the random operator is sampled and a component of the feasible set is randomly chosen. Such a sequential scheme is well suited for large-scale online and distributed learning. First, we assume that the VI is weak sharp. We provide asymptotic convergence, infeasibility rate of O (1/ k ) in terms of the squared distance to the feasible set, and solvability rate of O ( 1 / k ) in terms of the distance to the solution set for a bounded or unbounded set. Then, we assume just a monotone operator and introduce an explicit iterative Tykhonov regularization to the method. We consider Cartesian VIs so as to encompass the distributed solution of multiagent problems under a limited coordination. We provide asymptotic convergence, infeasibility rate of O (1/ k ) in terms of the squared distance to the feasible set and, in the case of a compact feasible set (with possibly unbounded components), we obtain a near-optimal solvability convergence rate of O ( k δ / k ) in terms of the dual gap function for any small δ ∈ ( 0 , 1 2 ) .

Suggested Citation

  • Alfredo N. Iusem & Alejandro Jofré & Philip Thompson, 2019. "Incremental Constraint Projection Methods for Monotone Stochastic Variational Inequalities," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 236-263, February.
  • Handle: RePEc:inm:ormoor:v:44:y:2019:i:1:p:236-263
    DOI: 10.1287/moor.2017.0922
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    References listed on IDEAS

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    1. Alan J. King & R. Tyrrell Rockafellar, 1993. "Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 148-162, February.
    2. J. Bello Cruz & A. Iusem, 2010. "Convergence of direct methods for paramonotone variational inequalities," Computational Optimization and Applications, Springer, vol. 46(2), pages 247-263, June.
    3. Bello Cruz, J.Y. & Iusem, A.N., 2015. "Full convergence of an approximate projection method for nonsmooth variational inequalities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 114(C), pages 2-13.
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    Cited by:

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    3. R. Díaz Millán & O. P. Ferreira & L. F. Prudente, 2021. "Alternating conditional gradient method for convex feasibility problems," Computational Optimization and Applications, Springer, vol. 80(1), pages 245-269, September.

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