Level-set methods for convex optimization

TL;DR

This paper introduces a level-set method for convex optimization that transforms complex problems into a sequence of parametric problems, enabling efficient solutions even with complicated constraints.

Contribution

It proposes a novel approach that swaps the roles of objective and constraints, using a zero-finding procedure with inexact evaluations to solve convex problems more effectively.

Findings

Applicable to low-rank semidefinite optimization

Effective for sparse optimization problems

Works with generalized linear models for inference

Abstract

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 and constraint functions, and instead approximately solves a sequence of parametric level-set problems. A zero-finding procedure, based on inexact function evaluations and possibly inexact derivative information, leads to an efficient solution scheme 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 generalized linear models for inference.