Augmented Lagrangian Method for Mathematical Programs with Second-Order Cone Complementarity Constraints
This paper investigates mathematical programs with second-order cone complementarity constraints (SOCMPCCs), which extend classical mathematical programs with complementarity constraints (MPCCs) by incorporating second-order cone structures. SOCMPCCs present significant theoretical and computational challenges, primarily due to the failure of standard constraint qualifications (such as Robinson's constraint qualification) at all feasible points. This difficulty hinders the direct application of classical nonlinear programming theories and algorithms. Motivated by the success of the augmented Lagrangian method (ALM) in solving MPCCs, we explore its extension to SOCMPCCs. The ALM, known for its matrix-free implementation and strong local convergence properties, is well suited for handling the intricate interplay between complementarity and second-order cone constraints. In this paper, we propose a tailored ALM algorithm framework for SOCMPCCs and establish its feasibility and convergence properties. We show that, under bounded ALM penalty parameters or bounded augmented Lagrangian functions, the generated sequence converges to feasible points of the SOCMPCC. Furthermore, under feasibility and additional SOCMPCC-nondegeneracy condition, we prove convergence to K-stationary points, which constitute a fundamental optimality condition for SOCMPCCs. Numerical experiments, including both illustrative examples and high-dimensional problems, are conducted to demonstrate the effectiveness and practical applicability of the proposed algorithm in addressing the challenges inherent in SOCMPCCs.