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

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OPTIMIZATION LETTERS 2023年第8期17卷 1957-1978页

作者： Erturk, Muzeyyen Salkim, Ahmet Adiyaman Univ Dept Math TR-02040 Adiyaman Turkiye

In this article, we use the superiorization methodology to investigate the bounded perturbations resilience of the gradient projection algorithm proposed in Erturk et al. (J Nonlinear Convex Anal 21(4):943-951, 2020) for solving the convex minimization problem in Hilbert space setting. We obtain that the perturbed version of this gradient projection algorithm converges weakly to a solution of the convex minimization problem just like itself. We support our conclusion with an example in an infinitely dimensional Hilbert space. We also show that the superiorization methodology can be applied to the split feasibility and the inverse linear problems with the help of the perturbed gradient projection algorithm.

关键词： Bounded perturbation resilience Superiorization gradient projection algorithm Convex minimization problems Linear inverse problems Split feasibility problems

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A New gradient projection algorithm for Convex Minimization Problem and its Application to Split Feasibility Problem

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VIETNAM JOURNAL OF MATHEMATICS 2022年第1期50卷 29-44页

作者： Erturk, Muzeyyen Kizmaz, Asiye Adiyaman Univ Dept Math Adiyaman Turkey

In this paper, we study convergence analysis of a new gradient projection algorithm for solving convex minimization problems in Hilbert spaces. We observe that the proposed gradient projection algorithm weakly converges to a minimum of convex function f which is defined from a closed and convex subset of a Hilbert space to Double-struck capital R. Also, we give a nontrivial example to illustrate our result in an infinite dimensional Hilbert space. We apply our result to solve the split feasibility problem.

关键词： gradient projection algorithm Convex optimization problem Fixed point Split feasibility problem

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The gradient projection algorithm for Smooth Sets and Functions in Nonconvex Case

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SET-VALUED AND VARIATIONAL ANALYSIS 2021年第2期29卷 341-360页

作者： Balashov, Maxim V. Russian Acad Sci VA Trapeznikov Inst Control Sci 65 Profsoyuznaya St Moscow 117997 Russia

We consider the problem of minimization for a function with Lipschitz continuous gradient on a proximally smooth and smooth manifold in a finite dimensional Euclidean space. We consider the Lezanski-Polyak-Lojasiewicz (LPL) conditions in this problem of constrained optimization. We prove that the gradient projection algorithm for the problem converges with a linear rate when the LPL condition holds.

关键词： Lipschitz continuous gradient Proximal smoothness gradient projection algorithm Metric projection Nonconvex extremal problem Lezanski-Polyak-Lojasiewicz condition

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The gradient projection algorithm for a proximally smooth set and a function with Lipschitz continuous gradient

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SBORNIK MATHEMATICS 2020年第4期211卷 481-504页

作者： Balashov, M. V. Russian Acad Sci VA Trapeznikov Inst Control Sci Moscow Russia

We consider the minimization problem for a nonconvex function with Lipschitz continuous gradient on a proximally smooth (possibly nonconvex) subset of a finite-dimensional Euclidean space. We introduce the error bound condition with exponent alpha is an element of(0, 1] for the gradient mapping. Under this condition, it is shown that the standard gradient projection algorithm converges to a solution of the problem linearly or sublinearly, depending on the value of the exponent alpha. This paper is theoretical. Bibliography: 23 titles.

关键词： gradient projection algorithm gradient mapping error bound condition proximal smoothness nonconvex extremal problem

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Linear convergence of gradient projection algorithm for split equality problems

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OPTIMIZATION 2018年第12期67卷 2347-2358页

作者： Shi, Luo-Yi Ansari, Qamrul Hasan Yao, Jen-Chih Wen, Ching-Feng Tianjin Polytech Univ Dept Math Tianjin Peoples R China Aligarh Muslim Univ Dept Math Aligarh Uttar Pradesh India King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran Saudi Arabia China Med Univ Ctr Gen Educ Taichung Taiwan Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung 80708 Taiwan Kaohsiung Med Univ Res Ctr Nonlinear Anal & Optimizat Kaohsiung 80708 Taiwan Kaohsiung Med Univ Hosp Dept Med Res Kaohsiung 80708 Taiwan

In this paper, we consider the varying stepsize gradient projection algorithm (GPA) for solving the split equality problem (SEP) in Hilbert spaces, and study its linear convergence. In particular, we introduce a notion of bounded linear regularity property for the SEP, and use it to establish the linear convergence property for the varying stepsize GPA. We provide some mild sufficient conditions to ensure the bounded linear regularity property, and then conclude the linear convergence rate of the varying stepsize GPA. To the best of our knowledge, this is the first work to study the linear convergence for the SEP.

关键词： Split equality problems gradient projection algorithm CQ algorithm linear convergence

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THE gradient projection algorithm WITH ADAPTIVE MUTATION STEP LENGTH FOR NON-PROBABILISTIC RELIABILITY INDEX

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TEHNICKI VJESNIK-TECHNICAL GAZETTE 2017年第1期24卷 53-62页

作者： Tang, Zhaoping Qin, Jin Sun, Jianping Geng, Biao Cent S Univ Sch Traff & Transportat Engn Changsha Hunan Peoples R China East China Jiao Tong Univ Sch Informat Engn 808 Shuanggang East Ave Nanchang 330013 Jiangxi Peoples R China Cent S Univ Sch Railway 68 Shaoshan South Rd Changsha 410075 Hunan Peoples R China East China Jiao Tong Univ Sch Railway Transportat 808 Shuanggang East Ave Nanchang 330013 Jiangxi Peoples R China

Aiming at the problems of selection parameter step-size and premature convergence that occurred when searching the local area in the optimal design of adaptive gradient projection algorithm in this paper, adaptive variable step-size mechanism strategy and adaptive variable step-size mechanism were established. They were introduced into the gradient projection algorithm, and were used to control iteration step length. Through the examples of non-probabilistic reliability index, it can be showed that the method could quickly and accurately calculate the reliability index when the model had multiple variables and complex limit state function. To compare and contrast this algorithm with the simple gradient projection algorithm, this algorithm is not sensitive to the initial point position. And it not only takes into account both local performance and global search ability, but also has fast convergence speed and high precision. So it is an efficient and practical optimization algorithm.

关键词： adaptive gradient projection algorithm mutation step length non-probabilistic reliability

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INCREMENTAL gradient projection algorithm FOR CONSTRAINED COMPOSITE MINIMIZATION PROBLEMS

JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS

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JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS 2017年第2期1卷 253-264页

作者： Shi, L. Y. Ansari, Q. H. Wen, C. F. Yao, J-C Tianjin Polytech Univ Dept Math Tianjin 300387 Peoples R China Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India Kaohsiung Med Univ Res Ctr Nonlinear Anal & Optimizat Kaohsiung 80424 Taiwan China Med Univ Ctr Gen Educ Taichung 40402 Taiwan

In this paper, we propose an incremental gradient projection algorithm for solving a minimization problem over the intersection of a finite family of closed convex subsets of a Hilbert space where the objective function is the sum of component functions. This algorithm is parameterized by a single nonnegative constant i mu. If mu = 0, then the proposed algorithm reduces to the classical incremental gradient method. The weak convergence of the sequence generated by the proposed algorithm is studied if the step size is chosen appropriately. Furthermore, in the special case of constrained least squares problem, the sequence generated by the proposed algorithm is proved to be convergent strongly to a solution of the constrained least squares problem under less requirements for the step size.

关键词： Composite minimization problem gradient projection algorithm Constrained least squares problem

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A novel flow update policy in solving traffic assignment problems: Successive over relaxation iteration method

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TRANSPORTATION RESEARCH PART E-LOGISTICS AND TRANSPORTATION REVIEW 2023年第1期174卷

作者： Zhang, Honggang Liu, Zhiyuan Wang, Jian Wu, Yunchi Southeast Univ Jiangsu Prov Collaborat Innovat Ctr Modern Urban T Sch Transportat Jiangsu Key Lab Urban ITS Nanjing 211189 Peoples R China

This paper presents a novel flow update policy, namely the successive over relaxation (SOR) iteration method, which can be implemented in traffic assignment algorithms. Most existing so-lution algorithms for the user equilibrium traffic assignment problem (UE-TAP) mainly use two flow update policies: Jacobi and Gauss-Seidel iteration methods. The proposed flow update policy SOR can be a more efficient replacement. Following the path-based gradient projection (GP) algorithm, we developed a new method GP-SOR for the UE-TAP. This study first provides the complete procedure of applying the GP-SOR algorithm to solve the UE-TAP. Subsequently, a few properties of the proposed method are rigorously proven. However, empirical tests of the GP-SOR algorithm demonstrate serious oscillations and poor convergence. To cope with this problem, the Armijo Rule is employed to determine the relaxation factor, which substantially improves the convergence of GP-SOR algorithm. The preliminary numerical examples show that the GP-SOR algorithm has speedier convergence compared with the known alternatives, which is reflected by the evident reduction of the computing time and the number of iterations.

关键词： User equilibrium Successive over relaxation iteration method gradient projection algorithm

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Self-adaptive gradient projection algorithms for variational inequalities involving non-Lipschitz continuous operators

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NUMERICAL algorithmS 2019年第3期81卷 983-1001页

作者： Pham Ky Anh Nguyen The Vinh Vietnam Natl Univ Coll Sci 334 Nguyen Trai Hanoi Vietnam Univ Transport & Commun Dept Math 3 Cau Giay St Hanoi Vietnam

In this paper, we introduce a self-adaptive inertial gradient projection algorithm for solving monotone or strongly pseudomonotone variational inequalities in real Hilbert spaces. The algorithm is designed such that the stepsizes are dynamically chosen and its convergence is guaranteed without the Lipschitz continuity and the paramonotonicity of the underlying operator. We will show that the proposed algorithm yields strong convergence without being combined with the hybrid/viscosity or linesearch methods. Our results improve and develop previously discussed gradient projection-type algorithms by Khanh and Vuong (J. Global Optim. 58, 341-350 2014).

关键词： Variational inequality Monotone operator gradient projection algorithm Extragradient algorithm Subgradient extragradient algorithm Projected reflected gradient method Inertial-type algorithm

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Error bound conditions and convergence of optimization methods on smooth and proximally smooth manifolds

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OPTIMIZATION 2022年第3期71卷 711-735页

作者： Balashov, M., V Tremba, A. A. Russian Acad Sci VA Trapeznikov Inst Control Sci Moscow Russia

We analyse the convergence of the gradient projection algorithm, which is finalized with the Newton method, to a stationary point for the problem of nonconvex constrained optimization minx. S f (x) with a proximally smooth set S = {x is an element of R-n : g(x) = 0}, g : R-n -> R-m and a smooth function f. Wepropose new Error bound (EB) conditions for the gradient projection method which lead to the convergence domain of the Newton method. We prove that these EB conditions are typical for a wide class of optimization problems. It is possible to reach high convergence rate of the algorithm by switching to the Newton method.

关键词： Error bound condition gradient projection algorithm Newton's method nonconvex optimization proximal smoothness

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