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A Single-Phase, Proximal Path-Following Framework

Author

Listed:
  • Quoc Tran-Dinh

    (Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599)

  • Anastasios Kyrillidis

    (Department of Computer Science, Rice University, Houston, Texas 77005)

  • Volkan Cevher

    (Laboratory for Information and Inference Systems, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland)

Abstract

We propose a new proximal path-following framework for a class of constrained convex problems. We consider settings where the nonlinear—and possibly nonsmooth—objective part is endowed with a proximity operator, and the constraint set is equipped with a self-concordant barrier. Our approach relies on the following two main ideas. First, we reparameterize the optimality condition as an auxiliary problem, such that a good initial point is available; by doing so, a family of alternative paths toward the optimum is generated. Second, we combine the proximal operator with path-following ideas to design a single-phase, proximal path-following algorithm. We prove that our algorithm has the same worst-case iteration complexity bounds as in standard path-following methods from the literature but does not require an initial phase. Our framework also allows inexactness in the evaluation of proximal Newton directions, without sacrificing the worst-case iteration complexity. We demonstrate the merits of our algorithm via three numerical examples, where proximal operators play a key role.

Suggested Citation

  • Quoc Tran-Dinh & Anastasios Kyrillidis & Volkan Cevher, 2018. "A Single-Phase, Proximal Path-Following Framework," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1326-1347, November.
    • Handle: RePEc:inm:ormoor:v:43:y:2018:i:4:p:1326-1347
    DOI: 10.1287/moor.2017.0907
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Yu. E. Nesterov & M. J. Todd, 1997. "Self-Scaled Barriers and Interior-Point Methods for Convex Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 1-42, February.
    3. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    4. NESTEROV, Yu., 2006. "Constructing self-concordant barriers for convex cones," LIDAM Discussion Papers CORE 2006030, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. NESTEROV, Yurii, 2011. "Barrier subgradient method," LIDAM Reprints CORE 2359, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Deyi Liu & Quoc Tran-Dinh, 2020. "An Inexact Interior-Point Lagrangian Decomposition Algorithm with Inexact Oracles," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 903-926, June.
    2. Qihang Lin & Runchao Ma & Yangyang Xu, 2022. "Complexity of an inexact proximal-point penalty method for constrained smooth non-convex optimization," Computational Optimization and Applications, Springer, vol. 82(1), pages 175-224, May.

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