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Proximal extrapolated gradient methods for variational inequalities.

Yu Malitsky1

  • 1Institute for Computer Graphics and Vision, Graz University of Technology, Graz, Austria.

Optimization Methods & Software
|January 20, 2018
PubMed
Summary
This summary is machine-generated.

This study introduces new first-order methods for monotone variational inequalities. These methods offer improved convergence rates without needing Lipschitz continuity, showing promising experimental results.

Keywords:
47J2065K1065K1565Y2090C33convex optimizationergodic convergencelinesearchmonotone operatornonmonotone stepsizesproximal methodsvariational inequality

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Area of Science:

  • Optimization
  • Numerical Analysis
  • Applied Mathematics

Background:

  • Monotone variational inequalities (VIs) are fundamental in optimization and game theory.
  • Existing methods often require strong assumptions like Lipschitz continuity of the operator.
  • Efficient algorithms for large-scale VIs remain an active research area.

Purpose of the Study:

  • To develop novel first-order methods for solving monotone variational inequalities.
  • To design methods that relax the Lipschitz continuity requirement for the operator.
  • To establish convergence rates and explore applicability to composite minimization problems.

Main Methods:

  • Introduction of new first-order iterative schemes.
  • Utilization of a simple linesearch procedure using only local operator information.
  • Adaptation of methods for composite minimization structures.

Main Results:

  • The proposed methods do not require Lipschitz continuity of the operator.
  • A simple linesearch procedure is employed, requiring only operator values.
  • Ergodic convergence rates are established for all presented methods.
  • A modified method demonstrates effectiveness for composite minimization.

Conclusions:

  • The novel methods provide efficient and robust approaches for monotone variational inequalities.
  • The relaxation of Lipschitz continuity broadens the applicability of the techniques.
  • Preliminary numerical experiments indicate strong performance and potential for practical use.