Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational...

Area of Science:

• Numerical Analysis
• Optimization Theory

Background:

• Variational inequalities are fundamental in applied mathematics and optimization.
• Solving variational inequalities often requires knowledge of operator properties like Lipschitz constants and strong monotonicity coefficients.
• Existing iterative methods may necessitate these parameters a priori, limiting their practical application.

Purpose of the Study:

• Introduce a novel self-adaptive iterative algorithm for variational inequalities.
• Develop a method that does not require prior knowledge of the Lipschitz constant or strong monotonicity coefficient.
• Analyze the convergence properties and error estimates of the proposed algorithm.

Main Methods:

• A self-adaptive iterative algorithm is proposed for solving variational inequalities.
• The algorithm utilizes a step size rule that adapts during iteration.
• Theoretical analysis is employed to prove strong convergence and derive a posteriori error estimates.

Main Results:

• The algorithm demonstrates strong convergence for solving variational inequalities.
• A novel self-adaptive step size rule is introduced, requiring minimal computational overhead.
• Numerical results indicate convergence rates comparable to the gradient projection method.

Conclusions:

• The proposed self-adaptive algorithm offers an efficient and practical approach to solving variational inequalities.
• The method's ability to avoid a priori parameter estimation enhances its applicability.
• The algorithm shows promising performance, warranting further investigation and application in relevant fields.