Mathematical Programming

, Volume 174, Issue 1–2, pp 359–390 | Cite as

Level-set methods for convex optimization

  • Aleksandr Y. AravkinEmail author
  • James V. Burke
  • Dmitry Drusvyatskiy
  • Michael P. Friedlander
  • Scott Roy
Full Length Paper Series B


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.


First-order methods Root-finding Semidefinite and conic programming 

Mathematics Subject Classification

90C25 65K10 49M29 90-08 



The authors extend their sincere thanks to three anonymous referees who provided an extensive list of corrections and suggestions that helped us to streamline our presentation.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA
  3. 3.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada

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