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 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. ∗ Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA; sites.google.com/ site/saravkin/; Research supported by the Washington Research Foundation Data Science Professorship. † Department of Mathematics, University of Washington, Seattle, WA 98195, USA; www.math.washington.edu/ ~burke/; Research supported in part by the NSF award DMS-1514559. ‡ Department of Mathematics, University of Washington, Seattle, WA 98195, USA; www.math.washington.edu/ ~ddrusv/; Research supported by the AFOSR YIP award FA9550-15-1-0237. § Department of Mathematics, UC Davis, One Shields Ave, Davis, CA 95616; www.math.ucdavis.edu/~mpf/; Research supported by the ONR award N00014-16-1-2242. ¶ Department of Mathematics, University of Washington, Seattle, WA 98195, USA; Research supported in part by the AFOSR YIP award FA9550-15-1-0237. 1 ar X iv :1 60 2. 01 50 6v 1 [ m at h. O C ] 3 F eb 2 01 6