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Complex-valued proximal neural network method for solving complex-valued mixed variational inequalities.

Yuanhua Wang1, Siru Yin1, Jun Li1

  • 1School of Mathematical Sciences, China West Normal University, Sichuan Colleges and Universities Key Laboratory of Optimization Theory and Applications, Nanchong, Sichuan, 637009, China.

Neural Networks : the Official Journal of the International Neural Network Society
|April 8, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces a complex-valued proximal neural network (CPNN) for solving complex-valued mixed variational inequalities (CMVIs). The CPNN method demonstrates well-posedness, stability, and effective convergence for complex-domain problems.

Keywords:
Complex-valued mixed variational inequalitiesComplex-valued proximal mappingComplex-valued proximal neural networkExponential stabilityLinear convergence

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Neural Networks

Background:

  • Variational inequalities are crucial in optimization and game theory.
  • Solving complex-valued problems requires specialized numerical methods.
  • Existing methods may not efficiently handle complex-domain inequalities.

Purpose of the Study:

  • To propose a novel complex-valued proximal neural network (CPNN) for solving complex-valued mixed variational inequalities (CMVIs).
  • To analyze the theoretical properties of the proposed CPNN, including solution existence, uniqueness, and stability.
  • To demonstrate the practical effectiveness of the CPNN through numerical experiments.

Main Methods:

  • Developing a complex-valued proximal mapping and analyzing its nonexpansiveness.
  • Establishing existence, uniqueness, and error bounds for CMVIs under specific monotonicity and continuity conditions.
  • Utilizing Lyapunov analysis and the complex chain rule to prove stability and convergence properties of the CPNN.
  • Implementing and testing the CPNN on relevant numerical problems.

Main Results:

  • The nonexpansiveness of the complex-valued proximal mapping is established.
  • Existence and uniqueness of solutions for CMVIs are proven, along with an associated error bound.
  • Equilibrium points of the CPNN are shown to coincide with CMVI solutions.
  • Global exponential stability of continuous-time dynamics and linear convergence of discrete-time implementation are demonstrated.
  • Numerical experiments confirm the effectiveness of the proposed CPNN method.

Conclusions:

  • The proposed CPNN offers a robust and efficient approach for solving CMVIs directly in the complex domain.
  • The theoretical analysis confirms the stability and convergence of the network.
  • The method shows significant promise for applications requiring complex-valued optimization and analysis.