Focuses on a study which examined the modification of type approximate trust region methods via two curvilinear paths for unconstrained optimization. Properties of the curvilinear paths; Description of a method which ...Focuses on a study which examined the modification of type approximate trust region methods via two curvilinear paths for unconstrained optimization. Properties of the curvilinear paths; Description of a method which combines line search technique with an approximate trust region algorithm; Information on the convergence analysis; Details on the numerical experiments.展开更多
In this paper,we propose an improved trust region method for solving unconstrained optimization problems.Different with traditional trust region methods,our algorithm does not resolve the subproblem within the trust r...In this paper,we propose an improved trust region method for solving unconstrained optimization problems.Different with traditional trust region methods,our algorithm does not resolve the subproblem within the trust region centered at the current iteration point,but within an improved one centered at some point located in the direction of the negative gradient,while the current iteration point is on the boundary set.We prove the global convergence properties of the new improved trust region algorithm and give the computational results which demonstrate the effectiveness of our algorithm.展开更多
In this paper, we investigate the quadratic approximation methods. After studying the basic idea of simplex methods, we construct several new search directions by combining the local information progressively obtained...In this paper, we investigate the quadratic approximation methods. After studying the basic idea of simplex methods, we construct several new search directions by combining the local information progressively obtained during the iterates of the algorithm to form new subspaces. And the quadratic model is solved in the new subspaces. The motivation is to use the information disclosed by the former steps to construct more promising directions. For most tested problems, the number of functions evaluations have been reduced obviously through our algorithms.展开更多
In this paper, a new trust region algorithm for nonlinear equality constrained LC1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subprobiem with...In this paper, a new trust region algorithm for nonlinear equality constrained LC1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subprobiem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.展开更多
In this paper, an ODE-type trust region algorithm for solving a class of nonlinear complementarity problems is proposed. A feature of this algorithm is that only the solution of linear systems of equations is required...In this paper, an ODE-type trust region algorithm for solving a class of nonlinear complementarity problems is proposed. A feature of this algorithm is that only the solution of linear systems of equations is required at each iteration, thus avoiding the need for solving a quadratic subproblem with a trust region bound. Under some conditions, it is proven that this algorithm is globally and locally superlinear convergent. The limited numerical examples show its efficiency.展开更多
In this paper we modify approximate trust region methods via three precon ditional curvilinear paths for unconstrained optimization. To easily form preconditional curvilinear paths within the trust region subproblem, ...In this paper we modify approximate trust region methods via three precon ditional curvilinear paths for unconstrained optimization. To easily form preconditional curvilinear paths within the trust region subproblem, we employ the stable Bunch-Parlett factorization method of symmetric matrices and use the unit lower triangular matrix as a preconditioner of the optimal path and modified gradient path. In order to accelerate the preconditional conjugate gradient path, we use preconditioner to improve the eigenvalue distribution of Hessian matrix. Based on the trial steps produced by the trust region subproblem along the three curvilinear paths providing a direction of sufficient descent, we mix a strategy using both trust region and nonmonotonic line search techniques which switch to back tracking steps when a trial step is unacceptable. Theoretical analysis is given to prove that the proposed algorithms are globally convergent and have a local su-pcrlinear convergent rate under some reasonable conditions. The results of the numerical experiment are reported to show the effectiveness of the proposed algorithms.展开更多
This paper studied subspace properties of the Celis–Dennis–Tapia(CDT)subproblem that arises in some trust-region algorithms for equality constrained optimization.The analysis is an extension of that presented by Wa...This paper studied subspace properties of the Celis–Dennis–Tapia(CDT)subproblem that arises in some trust-region algorithms for equality constrained optimization.The analysis is an extension of that presented by Wang and Yuan(Numer.Math.104:241–269,2006)for the standard trust-region subproblem.Under suitable conditions,it is shown that the trial step obtained from the CDT subproblem is in the subspace spanned by all the gradient vectors of the objective function and of the constraints computed until the current iteration.Based on this observation,a subspace version of the Powell–Yuan trust-region algorithm is proposed for equality constrained optimization problems where the number of constraints is much lower than the number of variables. The convergence analysis is given and numerical results arealso reported.展开更多
In this paper, a new trust region algorithm for minimax optimization problems is proposed, which solves only one quadratic subproblem based on a new approximation model at each iteration. The approach is different wit...In this paper, a new trust region algorithm for minimax optimization problems is proposed, which solves only one quadratic subproblem based on a new approximation model at each iteration. The approach is different with the traditional algorithms that usually require to solve two quadratic subproblems. Moreover, to avoid Maratos effect, the nonmonotone strategy is employed. The analysis shows that, under standard conditions, the algorithm has global and superlinear convergence. Preliminary numerical experiments are conducted to show the effiency of the new method.展开更多
We present a stochastic trust-region model-based framework in which its radius is related to the probabilistic models.Especially,we propose a specific algorithm termed STRME,in which the trust-region radius depends li...We present a stochastic trust-region model-based framework in which its radius is related to the probabilistic models.Especially,we propose a specific algorithm termed STRME,in which the trust-region radius depends linearly on the gradient used to define the latest model.The complexity results of the STRME method in nonconvex,convex and strongly convex settings are presented,which match those of the existing algorithms based on probabilistic properties.In addition,several numerical experiments are carried out to reveal the benefits of the proposed methods compared to the existing stochastic trust-region methods and other relevant stochastic gradient methods.展开更多
基金the Chinese National Science Foundation Grant 10071050, the Science andTechnology Foundation of Shanghai Higher Education.
文摘Focuses on a study which examined the modification of type approximate trust region methods via two curvilinear paths for unconstrained optimization. Properties of the curvilinear paths; Description of a method which combines line search technique with an approximate trust region algorithm; Information on the convergence analysis; Details on the numerical experiments.
基金supported by National Natural Science Foundation of China(Grant Nos.60903088 and 11101115)the Natural Science Foundation of Hebei Province(Grant No.A2010000188)Doctoral Foundation of Hebei University(Grant No.2008136)
文摘In this paper,we propose an improved trust region method for solving unconstrained optimization problems.Different with traditional trust region methods,our algorithm does not resolve the subproblem within the trust region centered at the current iteration point,but within an improved one centered at some point located in the direction of the negative gradient,while the current iteration point is on the boundary set.We prove the global convergence properties of the new improved trust region algorithm and give the computational results which demonstrate the effectiveness of our algorithm.
基金This work was partially supported by the Doctoral Foundation of Hebei University(Grant No.Y2006084)the National Natural Science Foundation of China(Grant No.10231060)
文摘In this paper, we investigate the quadratic approximation methods. After studying the basic idea of simplex methods, we construct several new search directions by combining the local information progressively obtained during the iterates of the algorithm to form new subspaces. And the quadratic model is solved in the new subspaces. The motivation is to use the information disclosed by the former steps to construct more promising directions. For most tested problems, the number of functions evaluations have been reduced obviously through our algorithms.
文摘In this paper, a new trust region algorithm for nonlinear equality constrained LC1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subprobiem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.
基金Supported by the Natural Science Foundation of Hainan Province(80552)
文摘In this paper, an ODE-type trust region algorithm for solving a class of nonlinear complementarity problems is proposed. A feature of this algorithm is that only the solution of linear systems of equations is required at each iteration, thus avoiding the need for solving a quadratic subproblem with a trust region bound. Under some conditions, it is proven that this algorithm is globally and locally superlinear convergent. The limited numerical examples show its efficiency.
文摘In this paper we modify approximate trust region methods via three precon ditional curvilinear paths for unconstrained optimization. To easily form preconditional curvilinear paths within the trust region subproblem, we employ the stable Bunch-Parlett factorization method of symmetric matrices and use the unit lower triangular matrix as a preconditioner of the optimal path and modified gradient path. In order to accelerate the preconditional conjugate gradient path, we use preconditioner to improve the eigenvalue distribution of Hessian matrix. Based on the trial steps produced by the trust region subproblem along the three curvilinear paths providing a direction of sufficient descent, we mix a strategy using both trust region and nonmonotonic line search techniques which switch to back tracking steps when a trial step is unacceptable. Theoretical analysis is given to prove that the proposed algorithms are globally convergent and have a local su-pcrlinear convergent rate under some reasonable conditions. The results of the numerical experiment are reported to show the effectiveness of the proposed algorithms.
基金G.N.Grapiglia was supported by Coordination for the Improvement of Higher Education Personnel(CAPES),Brazil(Grant PGCI No.12347/12-4).J.Yuan was partially supported by Coordination for the Improvement of Higher Education Personnel(CAPES)and by the National Council for Scientific and Technological Development(CNPq),Brazil.Y.-x.Yuan was partially supported by Natural Science Foundation of China,China(Grant No.11331012)This work was carried out while the first author was visiting Institute of Computational Mathematics and Scientific/Engineering Computing of the Chinese Academy of Sciences.He would like to thank Professor Ya-xiang Yuan,Professor Yu-hong Dai,Dr.Xin Liu and Dr.Ya-feng Liu for their warm hospitality.The authors also are grateful to Dr.Wei Leng for his help in installing and configuring the CUTEr.Finally,the authors would like to thank the two referees for their helpful comments.
文摘This paper studied subspace properties of the Celis–Dennis–Tapia(CDT)subproblem that arises in some trust-region algorithms for equality constrained optimization.The analysis is an extension of that presented by Wang and Yuan(Numer.Math.104:241–269,2006)for the standard trust-region subproblem.Under suitable conditions,it is shown that the trial step obtained from the CDT subproblem is in the subspace spanned by all the gradient vectors of the objective function and of the constraints computed until the current iteration.Based on this observation,a subspace version of the Powell–Yuan trust-region algorithm is proposed for equality constrained optimization problems where the number of constraints is much lower than the number of variables. The convergence analysis is given and numerical results arealso reported.
文摘In this paper, a new trust region algorithm for minimax optimization problems is proposed, which solves only one quadratic subproblem based on a new approximation model at each iteration. The approach is different with the traditional algorithms that usually require to solve two quadratic subproblems. Moreover, to avoid Maratos effect, the nonmonotone strategy is employed. The analysis shows that, under standard conditions, the algorithm has global and superlinear convergence. Preliminary numerical experiments are conducted to show the effiency of the new method.
基金This research is partially supported by the National Natural Science Foundation of China 11331012 and 11688101.
文摘We present a stochastic trust-region model-based framework in which its radius is related to the probabilistic models.Especially,we propose a specific algorithm termed STRME,in which the trust-region radius depends linearly on the gradient used to define the latest model.The complexity results of the STRME method in nonconvex,convex and strongly convex settings are presented,which match those of the existing algorithms based on probabilistic properties.In addition,several numerical experiments are carried out to reveal the benefits of the proposed methods compared to the existing stochastic trust-region methods and other relevant stochastic gradient methods.
