Bounded Perturbation Resilience Research Articles

Scaled forward-backward algorithm and the modified superiorized version for solving the split monotone variational inclusion problem

In this paper, we propose an inexact scaled forward-backward algorithm to solve the split monotone variational inclusion problem in a real Hilbert space and prove the strong convergence of it under appropriate conditions. Based on this, we discuss the bounded perturbation resilience of the exact algorithm for introducing the corresponding superiorized version and the superiorization algorithm with restarted perturbations. The numerical experiments illustrate that the proposed algorithms perform well and the superiorization version with restarted perturbations has advantage in decreasing the number of the iterations.

Perturbation Resilience of Self-Adaptive Step-Size Algorithms for Solving Split Variational Inclusion Problems and their Applications

In this paper, two viscosity proximal type algorithms involving the superiorization method are presented for solving the split variational inclusion problems in real Hilbert spaces. Strong convergence theorems and bounded perturbation resilience analysis of the proposed algorithms are obtained under mild conditions. The split feasibility problems, the split minimization problems, and the variational inequality problems are concerned as the applications, and several numerical experiments are performed to show the efficiency and implementation of the proposed algorithms.

Bounded perturbation resilience of a regularized forward-reflected-backward splitting method for solving variational inclusion problems with applications

ABSTRACT The forward-reflected-backward splitting method recently introduced for solving variational inclusion problems involves just one forward evaluation and one backward evaluation of the monotone operator and the maximal monotone operator, respectively, per iteration. This structure gives it some advantage over the earlier proposed methods. However, it only provides weak convergence, in general. Our aim in this paper is to improve the forward-reflected-backward splitting method in order to obtain strong convergence. To this end, we first study a regularized variational inclusion problem of finding the zero of the sum of two monotone operators. We then propose a regularized forward-reflected-backward splitting method for approximating a solution to the problem and prove the strong convergence of our iterative scheme under some suitable assumptions on the parameters. Moreover, we show that our algorithm has the bounded perturbation resilience property. Furthermore, we apply our results to convex minimization, split feasibility, split variational inclusion, and image deblurring problems, and illustrate the performance of our algorithm with several numerical examples.

Superiorization and bounded perturbation resilience of a gradient projection algorithm solving the convex minimization problem

Superiorization and bounded perturbation resilience of a gradient projection algorithm solving the convex minimization problem

Bounded Perturbation Resilience of Two Modified Relaxed CQ Algorithms for the Multiple-Sets Split Feasibility Problem

In this paper, we present some modified relaxed CQ algorithms with different kinds of step size and perturbation to solve the Multiple-sets Split Feasibility Problem (MSSFP). Under mild assumptions, we establish weak convergence and prove the bounded perturbation resilience of the proposed algorithms in Hilbert spaces. Treating appropriate inertial terms as bounded perturbations, we construct the inertial acceleration versions of the corresponding algorithms. Finally, for the LASSO problem and three experimental examples, numerical computations are given to demonstrate the efficiency of the proposed algorithms and the validity of the inertial perturbation.

Bounded perturbation resilience of viscosity proximal algorithm for solving split variational inclusion problems with applications to compressed sensing and image recovery

Split variational inclusion problems are core and useful technique that connects many important problems in nonlinear analysis. This paper proposes a resolvent‐based superiorization method for solving split variational inclusion problems, and some strong convergence theorems are proved under some mild conditions. Superiorization iterative procedures are simple yet powerful method that interlaces two algorithmic procedures under the assumption that one of the algorithmic procedure called the basic algorithm is resilient to the disturbances of its iterates caused by the activities of the other algorithm called perturbations. The numerical results in this paper show that our results are improvements and extensions of the corresponding results announced by many others.

Superiorization methodology and perturbation resilience of inertial proximal gradient algorithm with application to signal recovery

In this paper, we construct a novel algorithm for solving non-smooth composite optimization problems. By using inertial technique, we propose a modified proximal gradient algorithm with outer perturbations, and under standard mild conditions, we obtain strong convergence results for finding a solution of composite optimization problem. Based on bounded perturbation resilience, we present our proposed algorithm with the superiorization method and apply it to image recovery problem. Finally, we provide the numerical experiments to show efficiency of the proposed algorithm and comparison with previously known algorithms in signal recovery.

Strong convergence of over-relaxed multi-parameter proximal scaled gradient algorithm and superiorization

ABSTRACT In this paper, we propose an over-relaxed proximal scaled gradient algorithm for solving the non-smooth composite optimization problem in Hilbert space. We prove the strong convergence and the bounded perturbation resilience of this algorithm under some mild conditions. We also list the superiorized version of the algorithm. We apply the algorithms to the problem to illustrate the performance and validity. The numerical examples show that the over-relaxed algorithm can achieve fewer iteration steps and less running times in comparison to the under-relaxed algorithm, and that the superiorization algorithm has the minimum iterative steps among the three algorithms.

Bounded perturbation resilience of generalized viscosity iterative algorithms for split variational inclusion problems

Bounded perturbation resilience of generalized viscosity iterative algorithms for split variational inclusion problems

Bounded perturbation resilience and superiorization techniques for a modified proximal gradient method

ABSTRACTIn this article, we combine the viscosity approximation method and proximal operator to propose modified proximal gradient methods for solving the unconstrained convex optimization problems. The bounded perturbation resilience of these methods is investigated in the general Hilbert space H. Under reasonable parameter conditions, we prove that our algorithms strongly converge to the unique solution of a variational inequality problem. Our results improve and extend the corresponding results reported by many authors recently.

Bounded Perturbation Resilience and Superiorization of Proximal Scaled Gradient Algorithm with Multi-Parameters

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.

Accelerating Two Projection Methods via Perturbations with Application to Intensity-Modulated Radiation Therapy

Constrained convex optimization problems arise naturally in many real-world applications. One strategy to solve them in an approximate way is to translate them into a sequence of convex feasibility problems via the recently developed level set scheme and then solve each feasibility problem using projection methods. However, if the problem is ill-conditioned, projection methods often show zigzagging behavior and therefore converge slowly. To address this issue, we exploit the bounded perturbation resilience of the projection methods and introduce two new perturbations which avoid zigzagging behavior. The first perturbation is in the spirit of $k$-step methods and uses gradient information from previous iterates. The second uses the approach of surrogate constraint methods combined with relaxed, averaged projections. We apply two different projection methods in the unperturbed version, as well as the two perturbed versions, to linear feasibility problems along with nonlinear optimization problems arising from intensity-modulated radiation therapy (IMRT) treatment planning. We demonstrate that for all the considered problems the perturbations can significantly accelerate the convergence of the projection methods and hence the overall procedure of the level set scheme. For the IMRT optimization problems the perturbed projection methods found an approximate solution up to 4 times faster than the unperturbed methods while at the same time achieving objective function values which were 0.5 to 5.1% lower.

Bounded perturbation resilience of a viscosity iterative method for split feasibility problems

Bounded perturbation resilience of a viscosity iterative method for split feasibility problems

Strong Convergence Theorems of Viscosity Iterative Algorithms for Split Common Fixed Point Problems

In this paper, we propose a viscosity approximation method to solve the split common fixed point problem and consider the bounded perturbation resilience of the proposed method in general Hilbert spaces. Under some mild conditions, we prove that our algorithms strongly converge to a solution of the split common fixed point problem, which is also the unique solution of the variational inequality problem. Finally, we show the convergence and effectiveness of the algorithms by two numerical examples.

Strong convergence and bounded perturbation resilience of a modified proximal gradient algorithm

The proximal gradient algorithm is an appealing approach in finding solutions of non-smooth composite optimization problems, which may only has weak convergence in the infinite-dimensional setting. In this paper, we introduce a modified proximal gradient algorithm with outer perturbations in Hilbert space and prove that the algorithm converges strongly to a solution of the composite optimization problem. We also discuss the bounded perturbation resilience of the basic algorithm of this iterative scheme and illustrate it with an application.

Convergence of projection and contraction algorithms with outer perturbations and their applications to sparse signals recovery

In this paper, we study the bounded perturbation resilience of projection and contraction algorithms for solving variational inequality (VI) problems in real Hilbert spaces. Under typical and standard assumptions of monotonicity and Lipschitz continuity of the VI’s associated mapping, convergence of the perturbed projection and contraction algorithms is proved. Based on the bounded perturbed resilience of projection and contraction algorithms, we present some inertial projection and contraction algorithms. In addition, we show that the perturbed algorithms converge at the rate of O(1 / t).

Bounded perturbation resilience of extragradient-type methods and their applications

In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving a variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the methods can also be considered. Moreover, once an algorithm is proved to be bounded perturbation resilience, superiorization can be used, and this allows flexibility in choosing the bounded perturbations in order to obtain a superior solution, as well explained in the paper. We also discuss some inertial extragradient methods. Under mild and standard assumptions of monotonicity and Lipschitz continuity of the VI’s associated mapping, convergence of the perturbed extragradient and subgradient extragradient methods is proved. In addition we show that the perturbed algorithms converge at the rate of O(1/t). Numerical illustrations are given to demonstrate the performances of the algorithms.

Bounded perturbation resilience and superiorization techniques for the projected scaled gradient method

Bounded perturbation resilience and superiorization techniques for the projected scaled gradient (PSG) method are studied under the general Hilbert space setting. Weak convergence results of the (superiorized) PSG method and its relaxed version are proved under the assumption that the errors be summable. It is also shown that the PSG method converges in a sublinear rate and can be accelerated to the convergence rate . Applications to linear inverse problems and split feasibility problems are discussed.

Perturbation resilience and superiorization methodology of averaged mappings

We first prove the bounded perturbation resilience for the successive fixed point algorithm of averaged mappings, which extends the string-averaging projection and block-iterative projection methods. We then apply the superiorization methodology to a constrained convex minimization problem where the constraint set is the intersection of fixed point sets of a finite family of averaged mappings.

Bounded perturbation resilience of the viscosity algorithm

In this article, we investigate the bounded perturbation resilience of the viscosity algorithm and propose the superiorized version of the viscosity algorithm. The convergence of the proposed algorithm is analyzed for a nonexpansive mapping. A modified viscosity algorithm and its convergence is presented.