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Related Concept Videos

Application of Nonlinear Inequalities01:29

Application of Nonlinear Inequalities

A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality:  can be factored as: This form makes it easier to identify the values that cause the expression to equal zero. In this case, the key values are 3...
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Gradient and Del Operator

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Related Experiment Videos

The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space.

Y Censor1, A Gibali, S Reich

  • 1Department of Mathematics, University of Haifa, Mt. Carmel, 31905 Haifa, Israel.

Journal of Optimization Theory and Applications
|April 15, 2011
PubMed
Summary
This summary is machine-generated.

We introduce a subgradient extragradient method for solving variational inequalities. A modified algorithm also finds fixed points for nonexpansive mappings, with weak convergence proven for both.

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Area of Science:

  • Optimization theory
  • Functional analysis
  • Numerical analysis

Background:

  • Variational inequalities are fundamental in modeling various problems in applied mathematics and economics.
  • Solving variational inequalities efficiently is crucial for many computational tasks.
  • Nonexpansive mappings play a significant role in fixed-point theory and its applications.

Purpose of the Study:

  • To develop and analyze a subgradient extragradient method for solving variational inequalities in Hilbert spaces.
  • To propose a modified algorithm that simultaneously addresses variational inequalities and fixed-point problems.
  • To establish theoretical convergence guarantees for the proposed algorithms.

Main Methods:

  • The study employs the subgradient extragradient method, a class of iterative algorithms for solving convex optimization and related problems.
  • A modified algorithm incorporates an additional step to handle the fixed-point constraint of a nonexpansive mapping.
  • Weak convergence analysis is utilized to demonstrate the behavior of the algorithms.

Main Results:

  • A novel subgradient extragradient algorithm is presented for solving variational inequalities.
  • A modified algorithm is introduced that converges to a common solution of a variational inequality and a fixed-point problem.
  • Weak convergence theorems are established for both the standard and modified algorithms, confirming their effectiveness.

Conclusions:

  • The proposed subgradient extragradient method provides an effective approach for solving variational inequalities.
  • The modified algorithm offers a unified framework for tackling problems involving both variational inequalities and nonexpansive mappings.
  • The established weak convergence results support the practical applicability of these algorithms in relevant domains.