Distributed Extra-Gradient Methods for Constrained Optimization over Multiplex Networks

This paper develops a distributed extra-gradient method for constrained resource allocation over multiplex networks. The proposed formulation represents resource-balance and multiplex coordination constraints through a unified monotone residual operator, leading to a projected predictor-corrector algorithm that uses local computations and multiplex-neighbor communication, together with aggregation-and-broadcast when resource constraints couple non-neighboring agents. A convergence analysis establishes convergence of the projected extra-gradient iteration under monotonicity, Lipschitz continuity, feasibility, exact residual evaluation, and a sufficient step-size condition. A local linear rate requires additional error-bound regularity and is not claimed from the standing assumptions alone. The formulation is further extended to strongly connected weight-balanced directed multiplex networks by exploiting the skew-symmetric structure of the associated coupling operator. Numerical experiments are conducted on a three-layer multienergy resource-allocation benchmark with seven nodes per layer. The proposed method converges for unweighted undirected, weighted undirected, and weight-balanced directed multiplex topologies. A normalized step-size study illustrates the effect of the algorithmic step size on convergence speed, while a network-size study characterizes the computational behavior as the multiplex dimension increases. The latter reveals an increase in the residual Lipschitz constant and a corresponding deterioration in convergence for larger instances, identifying conditioning of the residual operator as an important limitation of the present first-order formulation. The results demonstrate the applicability of the proposed extra-gradient framework to constrained distributed optimization over heterogeneous multiplex networks and provide a basis for extensions involving adaptive preconditioning, acceleration, unbalanced directed networks, and uncertain or time-varying multienergy systems.