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

This paper proposes a complex-valued proximal neural network (CPNN) method for solving complex-valued mixed variational inequalities (CMVIs) directly in the complex-domain. Under standard convexity assumptions, we establish the nonexpansiveness of the complex-valued proximal mapping. Based on this proximal formulation, we further establish the existence and uniqueness of solutions to the CMVIs and derive a corresponding error bound under the assumptions of θ-Φ strong pseudomonotonicity and Lipschitz continuity. We then prove that the equilibrium points of the proposed CPNN coincide with the solutions of the CMVIs, and that the trajectories of the network are well posed. By combining the complex chain rule with Lyapunov analysis, we establish global exponential stability of the continuous-time dynamics and prove linear convergence of its discrete-time implementation under suitable discretization conditions. Finally, numerical experiments demonstrate the effectiveness of the proposed method.