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

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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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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FAST AND EFFICIENT gradient projection algorithm FOR DYNAMIC GENERATION DISPATCHING

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IEE PROCEEDINGS-C GENERATION TRANSMISSION AND DISTRIBUTION 1989年第5期136卷 295-302页

作者： GRANELLI, GP MARANNINO, P MONTAGNA, M SILVESTRI, A ENTE NAZL ENERGIA ELETT CTR RIC AUTOMAI-20093 COLOGNO MONZESEITALY POLITECN MILAN DEPARTIMENTO ELETTROTECNI-20133 MILANITALY

A gradient projection algorithm is presented for the dynamic dispatch of thermal generations, based upon an active set strategy and exploiting second order information in determining the descent directions. The algorithm is adapted to the model of the dynamic dispatch problem which consists of the minimisation of a nonlinear objective function subject to quadratic equality constraints and linear inequality constraints. The envisaged algorithm can be applied to the dispatching of thermal units during the steep load pick-up and drop-down periods over short time intervals, and can also be used in the day-before scheduling after the unit commitment has been performed. Efficiency is demonstrated by tests on large-scale networks representative of the Italian EHV system. The limited CPU time requirements also suggest a possible on-line application in determining the constrained participation factors to be used in the Automatic Generation Control function.

关键词： dynamic generation dispatching gradient projection algorithm day-before scheduling drop-down periods load dispatching steep load pick-up Italian EHV system scheduling unit commitment Power system management, operation and economics

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A self-adaptive gradient projection algorithm for the nonadditive traffic equilibrium problem

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COMPUTERS & OPERATIONS RESEARCH 2012年第2期39卷 127-138页

作者： Chen, Anthony Zhou, Zhong Xu, Xiangdong Utah State Univ Dept Civil & Environm Engn Logan UT 84322 USA Citilabs Tallahassee FL 32303 USA

gradient projection (GP) algorithm has been shown as an efficient algorithm for solving the traditional traffic equilibrium problem with additive route costs. Recently, GP has been extended to solve the nonadditive traffic equilibrium problem (NaTEP), in which the cost incurred on each route is not just a simple sum of the link costs on that route. However, choosing an appropriate stepsize, which is not known a priori, is a critical issue in GP for solving the NaTEP. Inappropriate selection of the stepsize can significantly increase the computational burden, or even deteriorate the convergence. In this paper, a self-adaptive gradient projection (SAGP) algorithm is proposed. The self-adaptive scheme has the ability to automatically adjust the stepsize according to the information derived from previous iterations. Furthermore, the SAGP algorithm still retains the efficient flow update strategy that only requires a simple projection onto the nonnegative orthant. Numerical results are also provided to illustrate the efficiency and robustness of the proposed algorithm. Published by Elsevier Ltd.

关键词： Traffic equilibrium problem Nonadditive route cost gradient projection algorithm Self-adaptive scheme

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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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A linearly convergent dual-based gradient projection algorithm for quadratically constrained convex minimization

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MATHEMATICS OF OPERATIONS RESEARCH 2006年第2期31卷 398-417页

作者： Beck, Amir Teboulle, Marc Technion Israel Inst Technol Fac Ind Engn & Management IL-32000 Haifa Israel Tel Aviv Univ Sch Math Sci IL-69978 Tel Aviv Israel

This paper presents a new dual formulation for quadratically constrained convex programs. The special structure of the derived dual problem allows us to apply the gradient projection algorithm to produce a simple expl... 详细信息

关键词： quadratically constrained problems gradient projection algorithm convex minimization rate of convergence analysis

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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 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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