The augmented Lagrangian method can approximately solve convex optimization with least constraint violation

There are many important practical optimization problems whose feasible regions not known to be nonempty or not, and optimizers of the objective function with least constraint violation prefer found. A natural way for dealing these is extend nonlinear problem as one optimizing over set points violation. This leads study shifted problem. paper focuses on constrained convex The sufficient condition closedness shifts presented continuity properties optimal value solution mapping studied. Properties conjugate dual discussed through relations between function. solvability investigated. It shown that, if violated shift in domain subdifferential function, then this has an unbounded set. Under condition, optimality conditions established term augmented Lagrangian. that Lagrangian method sequence converges multipliers unbounded. Moreover, it proved able find approximate linear rate convergence under error bound condition. applied illustrative second-order cone numerical results verify our theoretical results.