Abstract
In this paper, a multi-parameter proximal scaled gradient algorithm with outer perturbations is presented in real Hilbert space. The strong convergence of the generated sequence is proved. The bounded perturbation resilience and the superiorized version of the original algorithm are also discussed. The validity and the comparison with the use or not of superiorization of the proposed algorithms were illustrated by solving the l 1 − l 2 problem.
Highlights
The superiorization method, which was introduced by Censor in 2010 [1], can solve a broad class of nonlinear constrained optimal problems that result from many practical problems such as computed tomography [2], medical image recovery [3,4], convex feasibility problems [5,6], inverse problems of radiation therapy [7] and so on, which generates an automatic procedure based on the fact that the basic algorithm has the property of bounded perturbation resilience so that it is expected to get lower values of the objective function
The proximal gradient method is one of the popular iterative methods used for solving problem (1), which has received a lot of attention in the recent past due to its fast theoretical convergence rates and strong practical performance
We have proposed a proximal scaled gradient algorithm with multi-parameters and studied the strong convergence of it in a real Hilbert space for solving a composite optimization problem
Summary
The superiorization method, which was introduced by Censor in 2010 [1], can solve a broad class of nonlinear constrained optimal problems that result from many practical problems such as computed tomography [2], medical image recovery [3,4], convex feasibility problems [5,6], inverse problems of radiation therapy [7] and so on, which generates an automatic procedure based on the fact that the basic algorithm has the property of bounded perturbation resilience so that it is expected to get lower values of the objective function. We study the bounded perturbation resilience property and the corresponding superiorization of a proximal scaled gradient algorithm with multi-parameters for solving the following non-smooth composite optimization problem of the form min[ f ( x ) + g( x )] =: min Φ( x ), x∈ H (1). Motivated by [13], Guo, Cui and Guo [23] discussed the proximal gradient algorithm with perturbations: xn+1 = proxλn g ( I − λn D ∇ f + e) xn They proved that the generated sequence { xn } converges weakly to the solution of problem (1). Guo and Cui [15] applied the convex combination of contraction operator and proximal gradient operator to obtain the strong convergence of the generated sequence and discussed the bounded perturbation resilience of the exact algorithm.
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