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An extension of the proximal point algorithm beyond convexity

Author

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  • Sorin-Mihai Grad

    (University of Vienna)

  • Felipe Lara

    (Universidad de Tarapacá)

Abstract

We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity.

Suggested Citation

  • Sorin-Mihai Grad & Felipe Lara, 2022. "An extension of the proximal point algorithm beyond convexity," Journal of Global Optimization, Springer, vol. 82(2), pages 313-329, February.
    • Handle: RePEc:spr:jglopt:v:82:y:2022:i:2:d:10.1007_s10898-021-01081-4
    DOI: 10.1007/s10898-021-01081-4
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    References listed on IDEAS

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    1. Vial, Jean-Philippe, 1982. "Strong convexity of sets and functions," Journal of Mathematical Economics, Elsevier, vol. 9(1-2), pages 187-205, January.
    2. VIAL, Jean-Philippe, 1982. "Strong convexity of sets and functions," LIDAM Reprints CORE 475, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Suliman Al-Homidan & Nicolas Hadjisavvas & Loai Shaalan, 2018. "Transformation of Quasiconvex Functions to Eliminate Local Minima," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 93-105, April.
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    6. VIAL, Jean-Philippe, 1983. "Strong and weak convexity of sets and functions," LIDAM Reprints CORE 529, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Papa Quiroz, E.A. & Mallma Ramirez, L. & Oliveira, P.R., 2015. "An inexact proximal method for quasiconvex minimization," European Journal of Operational Research, Elsevier, vol. 246(3), pages 721-729.
    8. Teemu Pennanen, 2002. "Local Convergence of the Proximal Point Algorithm and Multiplier Methods Without Monotonicity," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 170-191, February.
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    Cited by:

    1. A. Kabgani & F. Lara, 2023. "Semistrictly and neatly quasiconvex programming using lower global subdifferentials," Journal of Global Optimization, Springer, vol. 86(4), pages 845-865, August.
    2. S.-M. Grad & F. Lara & R. T. Marcavillaca, 2023. "Relaxed-inertial proximal point type algorithms for quasiconvex minimization," Journal of Global Optimization, Springer, vol. 85(3), pages 615-635, March.
    3. Chinedu Izuchukwu & Yekini Shehu & Chibueze C. Okeke, 2023. "Extension of forward-reflected-backward method to non-convex mixed variational inequalities," Journal of Global Optimization, Springer, vol. 86(1), pages 123-140, May.

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