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Bounded perturbation resilience of projected scaled gradient methods

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  • Wenma Jin
  • Yair Censor
  • Ming Jiang

Abstract

We investigate projected scaled gradient (PSG) methods for convex minimization problems. These methods perform a descent step along a diagonally scaled gradient direction followed by a feasibility regaining step via orthogonal projection onto the constraint set. This constitutes a generalized algorithmic structure that encompasses as special cases the gradient projection method, the projected Newton method, the projected Landweber-type methods and the generalized expectation-maximization (EM)-type methods. We prove the convergence of the PSG methods in the presence of bounded perturbations. This resilience to bounded perturbations is relevant to the ability to apply the recently developed superiorization methodology to PSG methods, in particular to the EM algorithm. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • Wenma Jin & Yair Censor & Ming Jiang, 2016. "Bounded perturbation resilience of projected scaled gradient methods," Computational Optimization and Applications, Springer, vol. 63(2), pages 365-392, March.
    • Handle: RePEc:spr:coopap:v:63:y:2016:i:2:p:365-392
    DOI: 10.1007/s10589-015-9777-x
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    References listed on IDEAS

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    1. Charles Byrne & Yair Censor, 2001. "Proximity Function Minimization Using Multiple Bregman Projections, with Applications to Split Feasibility and Kullback–Leibler Distance Minimization," Annals of Operations Research, Springer, vol. 105(1), pages 77-98, July.
    2. M. V. Solodov & S. K. Zavriev, 1998. "Error Stability Properties of Generalized Gradient-Type Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 663-680, September.
    3. Jong-Shi Pang, 1987. "A Posteriori Error Bounds for the Linearly-Constrained Variational Inequality Problem," Mathematics of Operations Research, INFORMS, vol. 12(3), pages 474-484, August.
    4. Yair Censor & Ran Davidi & Gabor T. Herman & Reinhard W. Schulte & Luba Tetruashvili, 2014. "Projected Subgradient Minimization Versus Superiorization," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 730-747, March.
    5. John W. Chinneck, 2008. "Feasibility and Infeasibility in Optimization," International Series in Operations Research and Management Science, Springer, number 978-0-387-74932-7, April.
    6. Yair Censor & Alexander Zaslavski, 2013. "Convergence and perturbation resilience of dynamic string-averaging projection methods," Computational Optimization and Applications, Springer, vol. 54(1), pages 65-76, January.
    7. M. V. Solodov, 1997. "Convergence Analysis of Perturbed Feasible Descent Methods," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 337-353, May.
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    Cited by:

    1. Q. L. Dong & Y. J. Cho & L. L. Zhong & Th. M. Rassias, 2018. "Inertial projection and contraction algorithms for variational inequalities," Journal of Global Optimization, Springer, vol. 70(3), pages 687-704, March.
    2. Peichao Duan & Xubang Zheng & Jing Zhao, 2018. "Strong Convergence Theorems of Viscosity Iterative Algorithms for Split Common Fixed Point Problems," Mathematics, MDPI, vol. 7(1), pages 1-14, December.
    3. Yanni Guo & Xiaozhi Zhao, 2019. "Bounded Perturbation Resilience and Superiorization of Proximal Scaled Gradient Algorithm with Multi-Parameters," Mathematics, MDPI, vol. 7(6), pages 1-14, June.

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