An Augmented Lagrangian Method for Conic Convex Programming

We propose a new first-order augmented Lagrangian algorithm ALCC for solving convex conic programs of the form min { ρ(x) + γ(x) : Ax− b ∈ K, x ∈ χ } , where ρ : Rn → R ∪ {+∞}, γ : Rn → R are closed, convex functions, and γ has a Lipschitz continuous gradient, A ∈ Rm×n, K ⊂ Rm is a closed convex cone, and χ ⊂ dom(ρ) is a “simple” convex compact set such that optimization problems of the form min{ρ(x) + ‖x − x̄‖2 : x ∈ χ} can be efficiently solved. We show that any limit point of the primal ALCC iterates is an optimal solution of the conic convex problem, and the dual ALCC iterates have a unique limit point that is a KarushKuhn-Tucker (KKT) point of the conic program. We also show that for any > 0, the primal ALCC iterates are -feasible and -optimal after O(log( −1)) iterations which require solving O( −1 log( −1)) problems of the form minx{ρ(x)+‖x−x̄‖2 : x ∈ χ}.