An Augmented Lagrangian Method for Conic Convex Programming

TL;DR

This paper introduces a new augmented Lagrangian algorithm for convex conic programming that guarantees convergence to optimal solutions and provides an efficient complexity bound for achieving epsilon-optimality.

Contribution

The paper develops the ALCC algorithm for convex conic programs, proving convergence and establishing an O(log(1/epsilon)) iteration complexity for epsilon-approximate solutions.

Findings

Limit points of primal iterates are optimal solutions.

Dual iterates converge to a KKT point.

Algorithm achieves epsilon-optimality in O(log(1/epsilon)) iterations.

Abstract

We propose a new first-order augmented Lagrangian algorithm ALCC for solving convex conic programs of the form min{rho(x)+gamma(x): Ax-b in K, x in chi}, where rho and gamma are closed convex functions, and gamma has a Lipschitz continuous gradient, A is mxn real matrix, K is a closed convex cone, and chi is a "simple" convex compact set such that optimization problems of the form min{rho(x)+|x-x0|_2^2: x in chi} can be efficiently solved for any given x0. 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 Karush-Kuhn-Tucker (KKT) point of the conic program. We also show that for any epsilon>0, the primal ALCC iterates are epsilon-feasible and epsilon optimal after O(log(1/epsilon)) iterations which require solving O(1/epsilon log(1/epsilon)) problems of the form min{rho(x)+|x-x0|_2^2: x in chi}.