First-Order Method For Convex Smooth Function Constrained Variational Inequalities

Abstract

Constrained Variational Inequality (CVI) is a general optimization framework used to model and solve complementarity and equilibrium problems. In this paper, we first present a novel Accelerated Constrained Operator Extrapolation (ACOE) method for solving single-constrained monotone variational inequality (sCMVI) problems. We demonstrate that ACOE significantly improving the convergence rate for solving sCMVI problems to O (1/k) in terms of both operator and constraint's first-order evaluations, matching the optimal performance of unconstrained VI algorithms. Subsequently, to tackle more complex multi-constrained monotone variational inequality (mCMVI) problems, we propose the Accelerated Constrained Operator Extrapolation-Sliding (ACOE-S) algorithm. ACOE-S adopts Accelerated Constrained Gradient Descent (ACGD) method, Operator Extrapolation (OE) method, along with Primal-Dual Hybrid Gradient (PDHG) method to reduce the overall convergence rate to O (1/k). It then utilizes the sliding technique to future reduce the number of first-order evaluations required for handling multiple constraints to O 1/k 2. Finally, we conducted numerical experiments to validate the efficiency and effectiveness of the proposed algorithms for large-scale games with coupling constraints.

Keywords: constrained variational inequalities, accelerated constrained gradient method, stochastic approximation