Asymptotic strong duality

Given a nonconvex and nonsmooth optimization problem, we define a family of "perturbed" Lagrangians, which induce well-behaved approximations of the dual problem. Our family of approximated problems is said to verify strong asymptotic duality when the optimal dual values of the perturbed problems approach the primal optimal value. Our perturbed Lagrangians can have the same order of smoothness as the functions of the original problem, a property not shared by the classical (unperturbed) augmented Lagrangian. Therefore our proposed scheme allows the use of efficient numerical methods for solving the perturbed dual problems. We establish general conditions under which strong asymptotic duality holds, and we relate the latter with both strong duality and lower semicontinuity of the perturbation function. We illustrate our perturbed duality scheme with two important examples: Constrained Nonsmooth Optimization and Nonlinear Semidefinite programming.