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An infeasible projection type algorithm for nonmonotone variational inequalities

Abstract

It is well known that the monotonicity of the underlying mapping of variational inequalities plays a central role in the convergence analysis. In this paper, we propose an infeasible projection algorithm (IPA for short) for nonmonotone variational inequalities. The next iteration point of IPA is generated by projecting a vector onto a half-space. Hence, the computational cost of computing the next iteration point of IPA is much less than the algorithm of Ye and He (Comput. Optim. Appl. 60, 141–150, 2015) (YH for short). Moreover, if the underlying mapping is Lipschitz continuous with its modulus is known, by taking suitable parameters, IPA requires only one projection onto the feasible set per iteration. The global convergence of IPA is obtained when the solution set of its dual variational inequalities is nonempty. Moreover, if in addition error bound holds, the convergence rate of IPA is Q-linear. IPA can be used for a class of quasimonotone variational inequality problems and a class of quasiconvex minimization problems. Comparing with YH and Algorithm 2 in Deng, Hu and Fang (Numer. Algor. 86, 191–221, 2021) (DHF for short) by solving high-dimensional nonmonotone variational inequalities, numerical experiments show that IPA is much more efficient than YH and DHF from CPU time point of view. Moreover, IPA is less dependent on the initial value than YH and DHF.

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Notes

  1. 1.

    From Lemma 2.8 (b) below or [35, Theorem 3.5.4] , we see that the global minimizer on C of a smooth quasiconvex function f belongs to SD with F = ∇f. Hence, our algorithm can be applied in a class of quasiconvex optimization problem.

  2. 2.

    For each k, we take step-size αk as a fixed positive number to avoid computing αk by (??).

  3. 3.

    Here, disxS is used to denote the distance of the output point to S.

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Funding

This work is partially supported by the National Science Foundation of China (11871359, 11871059 and 11801455) and is supported by scientific research projects of China West Normal University (20A024).

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Correspondence to Minglu Ye.

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Ye, M. An infeasible projection type algorithm for nonmonotone variational inequalities. Numer Algor (2021). https://doi.org/10.1007/s11075-021-01170-1

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Keywords

  • Variational inequalities
  • Projection algorithm
  • Infeasible
  • Quasimonotone
  • Nonmonotone

Mathematics Subject Classification (2010)

  • 47J20
  • 49J40
  • 90C33