On perturbed hybrid steepest descent method with minimization or superiorization for subdifferentiable functions

For finding the minimum value of differentiable functions over a nonempty closed convex subset of a Hilbert space, the hybrid steepest descent method (HSDM) can be applied. In this work, we study perturbed algorithms in line with a generalized HSDM and discuss how some selections of perturbations enable us to increase the convergence speed. When we specialize these results to constrained minimization then the perturbations become bounded perturbations used in the superiorization methodology (SM). We show usefulness of the SM in studying the constrained convex minimization problem for subdifferentiable functions and proceed with the study of the computational efficiency of the SM compared with the HSDM. In the computational experiment comparing the HSDM with superiorization, the latter seems to be advantageous for the specific experiment.

[1]  Hong-Kun Xu.  An Iterative Approach to Quadratic Optimization , 2003 .

[2]  A. Zaslavski.  Asymptotic behavior of two algorithms for solving common fixed point problems , 2017 .

[3]  I. Yamada.  The Hybrid Steepest Descent Method for the Variational Inequality Problem over the Intersection of Fixed Point Sets of Nonexpansive Mappings , 2001 .

[4]  G. Herman,et al.  Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction , 2012, Inverse problems.

[5]  E. Llorens-Fuster,et al.  The fixed point property for mappings admitting a center , 2007 .

[6]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[7]  Andrzej Cegielski,et al.  Superiorization with level control , 2017 .

[8]  Alexander J. Zaslavski,et al.  Numerical Optimization with Computational Errors , 2016 .

[9]  Hongjin He,et al.  Perturbation resilience and superiorization methodology of averaged mappings , 2017 .

[10]  Shahram Saeidi,et al.  Combination of the Hybrid Steepest-Descent Method and the Viscosity Approximation , 2014, J. Optim. Theory Appl..

[11]  Ran Davidi,et al.  Superiorization: An optimization heuristic for medical physics , 2012, Medical physics.

[12]  I. Yamada,et al.  Nonexpansiveness of a linearized augmented Lagrangian operator for hierarchical convex optimization , 2017 .

[13]  A. Cegielski,et al.  Strong convergence of a hybrid steepest descent method for the split common fixed point problem , 2016 .

[14]  I. Yamada.  A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems , 2002 .

[15]  I. Yamada,et al.  Hybrid Steepest Descent Method for Variational Inequality Problem over the Fixed Point Set of Certain Quasi-nonexpansive Mappings , 2005 .

[16]  Yair Censor,et al.  Superiorization and Perturbation Resilience of Algorithms: A Continuously Updated Bibliography , 2015, 1506.04219.

[17]  Hideaki Iiduka.  Three-term conjugate gradient method for the convex optimization problem over the fixed point set of a nonexpansive mapping , 2011, Appl. Math. Comput..

[18]  B. Halpern.  Fixed points of nonexpanding maps , 1967 .

[19]  Ran Davidi,et al.  Projected Subgradient Minimization Versus Superiorization , 2013, Journal of Optimization Theory and Applications.

[20]  Yair Censor,et al.  Weak and Strong Superiorization: Between Feasibility-Seeking and Minimization , 2014, 1410.0130.

[21]  Yair Censor,et al.  Derivative-free superiorization with component-wise perturbations , 2018, Numerical Algorithms.

[22]  A. Moudafi.  Viscosity Approximation Methods for Fixed-Points Problems , 2000 .

[23]  F. Giannessi,et al.  Variational Analysis and Applications , 2005 .

[24]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[25]  Yair Censor,et al.  Strict Fejér Monotonicity by Superiorization of Feasibility-Seeking Projection Methods , 2014, J. Optim. Theory Appl..

[26]  Yair Censor.  Can Linear Superiorization Be Useful for Linear Optimization Problems? , 2017, Inverse problems.

[27]  Hong-Kun Xu,et al.  Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities , 2003 .

[28]  Simeon Reich,et al.  Convergence properties of dynamic string-averaging projection methods in the presence of perturbations , 2017, Numerical Algorithms.

[29]  Ran Davidi,et al.  Perturbation resilience and superiorization of iterative algorithms , 2010, Inverse problems.