An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities

Area of Science:

• Nonlinear analysis
• Optimization theory
• Functional analysis

Background:

• Inverse variational inequality (IVI) problems are crucial in applied mathematics and optimization.
• Existing methods for solving IVIs have limitations, particularly for complex constraints.

Purpose of the Study:

• To establish the unique solvability of a class of IVIs under Lipschitz continuity and strong monotonicity.
• To propose and analyze novel iterative algorithms for solving these IVIs.
• To extend existing results and provide practical computational tools.

Main Methods:

• Proving the existence and uniqueness of solutions for IVIs using fixed-point theory and monotonicity arguments.
• Developing the alternating contraction projection method (ACPM) based on projection techniques.
• Introducing the alternating contraction relaxation projection method (ACRPM) to handle complex projection operators.
• Analyzing the strong convergence and convergence rates of the proposed algorithms.

Main Results:

• A unique solution is proven to exist for IVIs where the nonlinear operator is Lipschitz continuous and strongly monotone.
• The strong convergence of the ACPM is established, along with a convergence rate estimate.
• The ACRPM demonstrates strong convergence, offering a viable alternative when projections are computationally challenging.
• Numerical experiments validate the practicability and effectiveness of both ACPM and ACRPM.

Conclusions:

• The study significantly advances the understanding and solvability of inverse variational inequalities.
• The proposed ACPM and ACRPM algorithms provide efficient and robust methods for solving these problems.
• The findings extend and improve upon existing literature in the field of variational inequalities.