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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...
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A generalized gradient projection method based on a new working set for minimax optimization problems with inequality

Guodong Ma1, Yufeng Zhang2, Meixing Liu1

  • 1School of Mathematics and Statistics, Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, China.

Journal of Inequalities and Applications
|March 17, 2017
PubMed
Summary
This summary is machine-generated.

This study introduces a novel generalized gradient projection algorithm for minimax optimization. The new method simplifies computation and demonstrates global convergence, showing promise for complex optimization tasks.

Keywords:
generalized gradient projection methodglobal and strong convergenceinequality constraintsminimax optimization problems

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Area of Science:

  • Optimization Theory
  • Numerical Analysis
  • Mathematical Programming

Background:

  • Minimax optimization problems with inequality constraints are prevalent in various scientific and engineering fields.
  • Existing algorithms often face challenges with computational complexity and convergence guarantees.
  • Efficient and reliable methods are crucial for solving these challenging optimization problems.

Purpose of the Study:

  • To develop a new generalized gradient projection algorithm for minimax optimization problems.
  • To improve computational efficiency by simplifying the search direction calculation.
  • To ensure the algorithm's global and strong convergence properties.

Main Methods:

  • The algorithm combines working set identification with generalized gradient projection techniques.
  • A novel optimal identification function is proposed to define a new working set.
  • The search direction is generated using a single, explicit generalized gradient projection formula per iteration.

Main Results:

  • The proposed algorithm offers a simplified computational approach.
  • It achieves global and strong convergence under mild assumptions.
  • Numerical results indicate the algorithm's promising performance.

Conclusions:

  • The new generalized gradient projection algorithm is effective for minimax optimization with inequality constraints.
  • The method's simplicity and convergence properties make it a valuable tool.
  • Further research and application of this algorithm are warranted.