In order to find roots of maximal monotone operators, this paper introduces and studies the modified approximate proximal point algorithm with an error sequence {e k} such that || ek || \leqslant hk || xk - [(x)\tilde...In order to find roots of maximal monotone operators, this paper introduces and studies the modified approximate proximal point algorithm with an error sequence {e k} such that || ek || \leqslant hk || xk - [(x)\tilde]k ||\left\| { e^k } \right\| \leqslant \eta _k \left\| { x^k - \tilde x^k } \right\| with ?k = 0￥ ( hk - 1 ) < + ￥\sum\limits_{k = 0}^\infty {\left( {\eta _k - 1} \right)} and infk \geqslant 0 hk = m\geqslant 1\mathop {\inf }\limits_{k \geqslant 0} \eta _k = \mu \geqslant 1 . Here, the restrictions on {η k} are very different from the ones on {η k}, given by He et al (Science in China Ser. A, 2002, 32 (11): 1026–1032.) that supk \geqslant 0 hk = v < 1\mathop {\sup }\limits_{k \geqslant 0} \eta _k = v . Moreover, the characteristic conditions of the convergence of the modified approximate proximal point algorithm are presented by virtue of the new technique very different from the ones given by He et al.展开更多
We introduced a new class of fuzzy set-valued variational inclusions with (H,η)-monotone mappings. Using the resolvent operator method in Hilbert spaces, we suggested a new proximal point algorithm for finding approx...We introduced a new class of fuzzy set-valued variational inclusions with (H,η)-monotone mappings. Using the resolvent operator method in Hilbert spaces, we suggested a new proximal point algorithm for finding approximate solutions, which strongly converge to the exact solution of a fuzzy set-valued variational inclusion with (H,η)-monotone. The results improved and generalized the general quasi-variational inclusions with fuzzy set-valued mappings proposed by Jin and Tian Jin MM, Perturbed proximal point algorithm for general quasi-variational inclusions with fuzzy set-valued mappings, OR Transactions, 2005, 9(3): 31-38, (In Chinese); Tian YX, Generalized nonlinear implicit quasi-variational inclusions with fuzzy mappings, Computers & Mathematics with Applications, 2001, 42: 101-108.展开更多
Proximal point algorithms （PPA） are attractive methods for solving monotone variational inequalities （MVI）. Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approx...Proximal point algorithms （PPA） are attractive methods for solving monotone variational inequalities （MVI）. Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions ofPPA （APPA） are developed for practical applications. In this paper, we compare two APPA methods, both of which can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. By extending the general forward-backward splitting methods, we obtain Algorithm Ⅰ; in the same way, Algorithm Ⅱ is proposed by spreading the general extra-gradient methods. Our analysis explains theoretically why Algorithm Ⅱ usually outperforms Algorithm Ⅰ. For computation practice, we consider a class of MVI with a special structure, and choose the extending Algorithm Ⅱ to implement, which is inspired by the idea of Gauss-Seidel iteration method making full use of information about the latest iteration. And in particular, self-adaptive techniques are adopted to adjust relevant parameters for faster convergence. Finally, some numerical experiments are reported on the separated MVI. Numerical results showed that the extending Algorithm II is feasible and easy to implement with relatively low computation load.展开更多
The proximal point algorithm has many interesting applications,such as signal recovery,signal processing and others.In recent years,the proximal point method has been extended to Riemannian manifolds.The main advantag...The proximal point algorithm has many interesting applications,such as signal recovery,signal processing and others.In recent years,the proximal point method has been extended to Riemannian manifolds.The main advantages of these extensions are that nonconvex problems in classic sense may become geodesic convex by introducing an appropriate Riemannian metric,constrained optimization problems may be seen as unconstrained ones.In this paper,we propose an inexact proximal point algorithm for geodesic convex vector function on Hadamard manifolds.Under the assumption that the objective function is coercive,the sequence generated by this algorithm converges to a Pareto critical point.When the objective function is coercive and strictly geodesic convex,the sequence generated by this algorithm converges to a Pareto optimal point.Furthermore,under the weaker growth condition,we prove that the inexact proximal point algorithm has linear/superlinear convergence rate.展开更多
In this paper, a new class of over-relaxed proximal point algorithms for solving nonlinear operator equations with (A,η,m)-monotonicity framework in Hilbert spaces is introduced and studied. Further, by using the gen...In this paper, a new class of over-relaxed proximal point algorithms for solving nonlinear operator equations with (A,η,m)-monotonicity framework in Hilbert spaces is introduced and studied. Further, by using the generalized resolvent operator technique associated with the (A,η,m)-monotone operators, the approximation solvability of the operator equation problems and the convergence of iterative sequences generated by the algorithm are discussed. Our results improve and generalize the corresponding results in the literature.展开更多
In this paper,we present an analysis about the rate of convergence of an inexact proximal point algorithm to solve minimization problems for quasiconvex objective functions on Hadamard manifolds.We prove that under na...In this paper,we present an analysis about the rate of convergence of an inexact proximal point algorithm to solve minimization problems for quasiconvex objective functions on Hadamard manifolds.We prove that under natural assumptions the sequence generated by the algorithm converges linearly or superlinearly to a critical point of the problem.展开更多
In this paper,we propose a new analysis framework to study the linear convergence of relaxed operator splitting methods,which can be viewed as an extension of the classic Krasnosel'skii-Mann iteration and Banach-P...In this paper,we propose a new analysis framework to study the linear convergence of relaxed operator splitting methods,which can be viewed as an extension of the classic Krasnosel'skii-Mann iteration and Banach-Picard contraction.As applications,we derive the linear convergence of the generalized proximal point algorithm and the relaxed forward-backward splitting method in a simple and elegant way.展开更多
K. Nakajo and W. Takahashi in 2003 proved the strong convergence theorems for nonex-pansive mappings, nonexpansive semigroups, and proximal point algorithm for zero point of monotone operators in Hilbert spaces by usi...K. Nakajo and W. Takahashi in 2003 proved the strong convergence theorems for nonex-pansive mappings, nonexpansive semigroups, and proximal point algorithm for zero point of monotone operators in Hilbert spaces by using the hybrid method in mathematical programming. The purpose of this paper is to modify the hybrid iteration method of K. Nakajo and W. Takahashi through the monotone hybrid method, and to prove strong convergence theorems. The convergence rate of iteration process of the monotone hybrid method is faster than that of the iteration process of the hybrid method of K. Nakajo and W. Takahashi. In the proofs in this article, Cauchy sequence method is used to avoid the use of the demiclosedness principle and Opial＇s condition.展开更多
A unified efficient algorithm framework of proximal-based decomposition methods has been proposed for monotone variational inequalities in 2012,while only global convergence is proved at the same time.In this paper,we...A unified efficient algorithm framework of proximal-based decomposition methods has been proposed for monotone variational inequalities in 2012,while only global convergence is proved at the same time.In this paper,we give a unified proof on theO(1/t)iteration complexity,together with the linear convergence rate for this kind of proximal-based decomposition methods.Besides theε-optimal iteration complexity result defined by variational inequality,the non-ergodic relative error of adjacent iteration points is also proved to decrease in the same order.Further,the linear convergence rate of this algorithm framework can be constructed based on some special variational inequality properties,without necessary strong monotone conditions.展开更多
Proximal point algorithm(PPA)is a useful algorithm framework and has good convergence properties.Themain difficulty is that the subproblems usually only have iterative solutions.In this paper,we propose an inexact cus...Proximal point algorithm(PPA)is a useful algorithm framework and has good convergence properties.Themain difficulty is that the subproblems usually only have iterative solutions.In this paper,we propose an inexact customized PPA framework for twoblock separable convex optimization problem with linear constraint.We design two types of inexact error criteria for the subproblems.The first one is absolutely summable error criterion,under which both subproblems can be solved inexactly.When one of the two subproblems is easily solved,we propose another novel error criterion which is easier to implement,namely relative error criterion.The relative error criterion only involves one parameter,which is more implementable.We establish the global convergence and sub-linear convergence rate in ergodic sense for the proposed algorithms.The numerical experiments on LASSO regression problems and total variation-based image denoising problem illustrate that our new algorithms outperform the corresponding exact algorithms.展开更多
The problem concerned in this paper is the set-valued equation 0 ∈ T(z) where T is a maximal monotone operator. For given xk and βk >: 0, some existing approximate proximal point algorithms take $x^{k + 1} = \til...The problem concerned in this paper is the set-valued equation 0 ∈ T(z) where T is a maximal monotone operator. For given xk and βk >: 0, some existing approximate proximal point algorithms take $x^{k + 1} = \tilde x^k $ such that $$x^k + e^k \in \tilde x^k + \beta _k T(\tilde x^k ) and \left\| {e^k } \right\| \leqslant \eta _k \left\| {x^k - \tilde x^k } \right\|,$$ where ?k is a non-negative summable sequence. Instead of $x^{k + 1} = \tilde x^k $ , the new iterate of the proposing method is given by $$x^{k + 1} = P_\Omega [\tilde x^k - e^k ],$$ where Ω is the domain of T and PΩ(·) denotes the projection on Ω. The convergence is proved under a significantly relaxed restriction supK>0 ηKη1.展开更多
Sparse signal recovery is a topic of considerable interest,and the literature in this field is already quite immense.Many problems that arise in sparse signal recovery can be generalized as a convex programming with l...Sparse signal recovery is a topic of considerable interest,and the literature in this field is already quite immense.Many problems that arise in sparse signal recovery can be generalized as a convex programming with linear conic constraints.In this paper,we present a new proximal point algorithm(PPA) termed as relaxed-PPA(RPPA) contraction method,for solving this common convex programming.More precisely,we first reformulate the convex programming into an equivalent variational inequality(VI),and then efficiently explore its inner structure.In each step,our method relaxes the VI-subproblem to a tractable one,which can be solved much more efficiently than the original VI.Under mild conditions,the convergence of the proposed method is proved.Experiments with l1 analysis show that RPPA is a computationally efficient algorithm and compares favorably with the recently proposed state-of-the-art algorithms.展开更多
In this paper we study the proximal point algorithm (PPA) based predictioncorrection (PC) methods for monotone variational inequalities. Each iteration of these methods consists of a prediction and a correction. The p...In this paper we study the proximal point algorithm (PPA) based predictioncorrection (PC) methods for monotone variational inequalities. Each iteration of these methods consists of a prediction and a correction. The predictors are produced by inexact PPA steps. The new iterates are then updated by a correction using the PPA formula. We present two profit functions which serve two purposes: First we show that the profit functions are tight lower bounds of the improvements obtained in each iteration. Based on this conclusion we obtain the convergence inexactness restrictions for the prediction step. Second we show that the profit functions are quadratically dependent upon the step lengths, thus the optimal step lengths are obtained in the correction step. In the last part of the paper we compare the strengths of different methods based on their inexactness restrictions.展开更多
In this paper, a bundle modification strategy is proposed for nonsmooth convex constrained min- imization problems. As a result, a new feasible point bundle method is presented by applying this strategy. Whenever the ...In this paper, a bundle modification strategy is proposed for nonsmooth convex constrained min- imization problems. As a result, a new feasible point bundle method is presented by applying this strategy. Whenever the stability center is updated, some points in the bundle will be substituted by new ones which have lower objective values and/or constraint values, aiming at getting a better bundle. The method generates feasible serious iterates on which the objective function is monotonically decreasing. Global convergence of the algorithm is established, and some preliminary numerical results show that our method performs better than the standard feasible point bundle method.展开更多
In this paper,we present a family of gradient projection method with arbitrary initialpoint.The formula of search direction in the method is unitary.The convergent conditions ofthe method are given.When the initial po...In this paper,we present a family of gradient projection method with arbitrary initialpoint.The formula of search direction in the method is unitary.The convergent conditions ofthe method are given.When the initial point is feasible,the family of the method contains severalknown algorithms.When the initial point is infeasible,the method is exactly that given in[6].Finally,we give a new method which has global convergence property.展开更多
基金Supported both by the Teaching and Research Award Fund for Outstanding Young Teachers inHigher Educational Institutions of MOEChinaand by the Dawn Program Fund in Shanghai
文摘In order to find roots of maximal monotone operators, this paper introduces and studies the modified approximate proximal point algorithm with an error sequence {e k} such that || ek || \leqslant hk || xk - [(x)\tilde]k ||\left\| { e^k } \right\| \leqslant \eta _k \left\| { x^k - \tilde x^k } \right\| with ?k = 0￥ ( hk - 1 ) < + ￥\sum\limits_{k = 0}^\infty {\left( {\eta _k - 1} \right)} and infk \geqslant 0 hk = m\geqslant 1\mathop {\inf }\limits_{k \geqslant 0} \eta _k = \mu \geqslant 1 . Here, the restrictions on {η k} are very different from the ones on {η k}, given by He et al (Science in China Ser. A, 2002, 32 (11): 1026–1032.) that supk \geqslant 0 hk = v < 1\mathop {\sup }\limits_{k \geqslant 0} \eta _k = v . Moreover, the characteristic conditions of the convergence of the modified approximate proximal point algorithm are presented by virtue of the new technique very different from the ones given by He et al.
基金the Natural Science Foundation of China (No. 10471151)the Educational Science Foundation of Chongqing (KJ051307).
文摘We introduced a new class of fuzzy set-valued variational inclusions with (H,η)-monotone mappings. Using the resolvent operator method in Hilbert spaces, we suggested a new proximal point algorithm for finding approximate solutions, which strongly converge to the exact solution of a fuzzy set-valued variational inclusion with (H,η)-monotone. The results improved and generalized the general quasi-variational inclusions with fuzzy set-valued mappings proposed by Jin and Tian Jin MM, Perturbed proximal point algorithm for general quasi-variational inclusions with fuzzy set-valued mappings, OR Transactions, 2005, 9(3): 31-38, (In Chinese); Tian YX, Generalized nonlinear implicit quasi-variational inclusions with fuzzy mappings, Computers & Mathematics with Applications, 2001, 42: 101-108.
基金Project (No. 1027054) supported by the National Natural Science Foundation of China
文摘Proximal point algorithms （PPA） are attractive methods for solving monotone variational inequalities （MVI）. Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions ofPPA （APPA） are developed for practical applications. In this paper, we compare two APPA methods, both of which can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. By extending the general forward-backward splitting methods, we obtain Algorithm Ⅰ; in the same way, Algorithm Ⅱ is proposed by spreading the general extra-gradient methods. Our analysis explains theoretically why Algorithm Ⅱ usually outperforms Algorithm Ⅰ. For computation practice, we consider a class of MVI with a special structure, and choose the extending Algorithm Ⅱ to implement, which is inspired by the idea of Gauss-Seidel iteration method making full use of information about the latest iteration. And in particular, self-adaptive techniques are adopted to adjust relevant parameters for faster convergence. Finally, some numerical experiments are reported on the separated MVI. Numerical results showed that the extending Algorithm II is feasible and easy to implement with relatively low computation load.
文摘The proximal point algorithm has many interesting applications,such as signal recovery,signal processing and others.In recent years,the proximal point method has been extended to Riemannian manifolds.The main advantages of these extensions are that nonconvex problems in classic sense may become geodesic convex by introducing an appropriate Riemannian metric,constrained optimization problems may be seen as unconstrained ones.In this paper,we propose an inexact proximal point algorithm for geodesic convex vector function on Hadamard manifolds.Under the assumption that the objective function is coercive,the sequence generated by this algorithm converges to a Pareto critical point.When the objective function is coercive and strictly geodesic convex,the sequence generated by this algorithm converges to a Pareto optimal point.Furthermore,under the weaker growth condition,we prove that the inexact proximal point algorithm has linear/superlinear convergence rate.
文摘In this paper, a new class of over-relaxed proximal point algorithms for solving nonlinear operator equations with (A,η,m)-monotonicity framework in Hilbert spaces is introduced and studied. Further, by using the generalized resolvent operator technique associated with the (A,η,m)-monotone operators, the approximation solvability of the operator equation problems and the convergence of iterative sequences generated by the algorithm are discussed. Our results improve and generalize the corresponding results in the literature.
基金Coordenação de Aperfeiçoamento de Pessoal de Nível Superior of the Federal University of Rio de Janeiro(UFRJ),Brazil.
文摘In this paper,we present an analysis about the rate of convergence of an inexact proximal point algorithm to solve minimization problems for quasiconvex objective functions on Hadamard manifolds.We prove that under natural assumptions the sequence generated by the algorithm converges linearly or superlinearly to a critical point of the problem.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1180145511871059+4 种基金11971238)China Postdoctoral Science Foundation(Grant Nos.2019M6634592020T130081)the Applied Basic Project of Sichuan Province(Grant No.2020YJ0111)the Open Project of Key Laboratory of School of Mathematical Sciences,Chongqing Normal University(Grant No.CSSXKFKTM202004)。
文摘In this paper,we propose a new analysis framework to study the linear convergence of relaxed operator splitting methods,which can be viewed as an extension of the classic Krasnosel'skii-Mann iteration and Banach-Picard contraction.As applications,we derive the linear convergence of the generalized proximal point algorithm and the relaxed forward-backward splitting method in a simple and elegant way.
基金This research is supported by the National Natural Science Foundation of China under Grant No.10771050
文摘K. Nakajo and W. Takahashi in 2003 proved the strong convergence theorems for nonex-pansive mappings, nonexpansive semigroups, and proximal point algorithm for zero point of monotone operators in Hilbert spaces by using the hybrid method in mathematical programming. The purpose of this paper is to modify the hybrid iteration method of K. Nakajo and W. Takahashi through the monotone hybrid method, and to prove strong convergence theorems. The convergence rate of iteration process of the monotone hybrid method is faster than that of the iteration process of the hybrid method of K. Nakajo and W. Takahashi. In the proofs in this article, Cauchy sequence method is used to avoid the use of the demiclosedness principle and Opial＇s condition.
基金The work was supported in part by the Shanghai Youth Science and Technology Talent Sail Plan(No.15YF1403400)the National Natural Science Foundation of China(No.61321064).
文摘A unified efficient algorithm framework of proximal-based decomposition methods has been proposed for monotone variational inequalities in 2012,while only global convergence is proved at the same time.In this paper,we give a unified proof on theO(1/t)iteration complexity,together with the linear convergence rate for this kind of proximal-based decomposition methods.Besides theε-optimal iteration complexity result defined by variational inequality,the non-ergodic relative error of adjacent iteration points is also proved to decrease in the same order.Further,the linear convergence rate of this algorithm framework can be constructed based on some special variational inequality properties,without necessary strong monotone conditions.
基金the National Natural Science Foundation of China(Nos.11971238 and 11871279)。
文摘Proximal point algorithm(PPA)is a useful algorithm framework and has good convergence properties.Themain difficulty is that the subproblems usually only have iterative solutions.In this paper,we propose an inexact customized PPA framework for twoblock separable convex optimization problem with linear constraint.We design two types of inexact error criteria for the subproblems.The first one is absolutely summable error criterion,under which both subproblems can be solved inexactly.When one of the two subproblems is easily solved,we propose another novel error criterion which is easier to implement,namely relative error criterion.The relative error criterion only involves one parameter,which is more implementable.We establish the global convergence and sub-linear convergence rate in ergodic sense for the proposed algorithms.The numerical experiments on LASSO regression problems and total variation-based image denoising problem illustrate that our new algorithms outperform the corresponding exact algorithms.
基金This work was supported by the National Natural Science Foundation of China(Grant No. 10271054), Natural Science Foundation of Jiangsu Province (Grant No. BK2002075) and grant FRG/00-01/11-63 of Hong Kong Baptist University.
文摘The problem concerned in this paper is the set-valued equation 0 ∈ T(z) where T is a maximal monotone operator. For given xk and βk >: 0, some existing approximate proximal point algorithms take $x^{k + 1} = \tilde x^k $ such that $$x^k + e^k \in \tilde x^k + \beta _k T(\tilde x^k ) and \left\| {e^k } \right\| \leqslant \eta _k \left\| {x^k - \tilde x^k } \right\|,$$ where ?k is a non-negative summable sequence. Instead of $x^{k + 1} = \tilde x^k $ , the new iterate of the proposing method is given by $$x^{k + 1} = P_\Omega [\tilde x^k - e^k ],$$ where Ω is the domain of T and PΩ(·) denotes the projection on Ω. The convergence is proved under a significantly relaxed restriction supK>0 ηKη1.
基金the National Natural Science Foundation of China(No.70901018)
文摘Sparse signal recovery is a topic of considerable interest,and the literature in this field is already quite immense.Many problems that arise in sparse signal recovery can be generalized as a convex programming with linear conic constraints.In this paper,we present a new proximal point algorithm(PPA) termed as relaxed-PPA(RPPA) contraction method,for solving this common convex programming.More precisely,we first reformulate the convex programming into an equivalent variational inequality(VI),and then efficiently explore its inner structure.In each step,our method relaxes the VI-subproblem to a tractable one,which can be solved much more efficiently than the original VI.Under mild conditions,the convergence of the proposed method is proved.Experiments with l1 analysis show that RPPA is a computationally efficient algorithm and compares favorably with the recently proposed state-of-the-art algorithms.
基金The author was supported by NSFC Grant 10271054MOEC grant 20020284027 and Jiangsur NSF grant BK20002075.
文摘In this paper we study the proximal point algorithm (PPA) based predictioncorrection (PC) methods for monotone variational inequalities. Each iteration of these methods consists of a prediction and a correction. The predictors are produced by inexact PPA steps. The new iterates are then updated by a correction using the PPA formula. We present two profit functions which serve two purposes: First we show that the profit functions are tight lower bounds of the improvements obtained in each iteration. Based on this conclusion we obtain the convergence inexactness restrictions for the prediction step. Second we show that the profit functions are quadratically dependent upon the step lengths, thus the optimal step lengths are obtained in the correction step. In the last part of the paper we compare the strengths of different methods based on their inexactness restrictions.
基金Project supported by the National Natural Science Foundation of China(11761013,11771383)Guangxi Natural Science Foundation(2013GXNSFAA019013,2014GXNSFFA118001,2016GXNSFDA380019)the Open Project of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing(2016CSOBDP0203)
文摘In this paper, a bundle modification strategy is proposed for nonsmooth convex constrained min- imization problems. As a result, a new feasible point bundle method is presented by applying this strategy. Whenever the stability center is updated, some points in the bundle will be substituted by new ones which have lower objective values and/or constraint values, aiming at getting a better bundle. The method generates feasible serious iterates on which the objective function is monotonically decreasing. Global convergence of the algorithm is established, and some preliminary numerical results show that our method performs better than the standard feasible point bundle method.
基金Project supported by the National Natural Science Foundation of China
文摘In this paper,we present a family of gradient projection method with arbitrary initialpoint.The formula of search direction in the method is unitary.The convergent conditions ofthe method are given.When the initial point is feasible,the family of the method contains severalknown algorithms.When the initial point is infeasible,the method is exactly that given in[6].Finally,we give a new method which has global convergence property.
