Level-set methods for convex optimization

Convex optimization problems arising in applications often have favorable objective functions and complicated constraints, thereby precluding first-order methods from being immediately applicable. We describe an approach that exchanges the roles of the objective with one of the constraint functions, and instead approximately solves a sequence of parametric level-set problems. Two Newton-like zero-finding procedures for nonsmooth convex functions, based on inexact evaluations and sensitivity information, are introduced. It is shown that they lead to efficient solution schemes for the original problem. We describe the theoretical and practical properties of this approach for a broad range of problems, including low-rank semidefinite optimization, sparse optimization, and gauge optimization.