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The Popov's Algorithm with Optimal Bounded Stepsize for Generalized Monotone Variational Inequalities

Nhung Hong Nguyen, Thanh Quoc Trinh, Phan Tu Vuong

Abstract

For solving constrained (pseudo)-monotone variational inequality, we prove that the upper bound of stepsize $\frac{1}{2L}$ established for the Popov's algorithm and the forward-reflected-backward algorithm is tight. For unconstrained case, we can enlarge the upper bound to $\frac{1}{\sqrt{3}L}$ and show that this upper bound is also tight. The convergence analysis is carried out by using a new Lyapunov-type function.

The Popov's Algorithm with Optimal Bounded Stepsize for Generalized Monotone Variational Inequalities

Abstract

For solving constrained (pseudo)-monotone variational inequality, we prove that the upper bound of stepsize established for the Popov's algorithm and the forward-reflected-backward algorithm is tight. For unconstrained case, we can enlarge the upper bound to and show that this upper bound is also tight. The convergence analysis is carried out by using a new Lyapunov-type function.
Paper Structure (5 sections, 4 theorems, 33 equations, 1 figure)

This paper contains 5 sections, 4 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. The tightness of stepsize upper bound $\frac{1}{2L}$ for constrained VI
  4. Larger stepsize and tight upper bound for unconstrained VI
  5. Conclusion

Key Result

Theorem 2.1

If $\lambda = \frac{1}{2L}$ then there exists a monotone and Lipchitz continuous operator $F$ and a closed convex set $K$ such that the Popov iterations PP do not converge to a solution.

Figures (1)

  • Figure :

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 2.3
  • ...and 1 more