In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by ad...In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.展开更多
A trust-region algorithm is presented for a nonlinear optimization problem of equality-constraints. The characterization of the algorithm is using inexact gradient information. Global convergence results are demonstra...A trust-region algorithm is presented for a nonlinear optimization problem of equality-constraints. The characterization of the algorithm is using inexact gradient information. Global convergence results are demonstrated where the gradient values are obeyed a simple relative error condition.展开更多
A conic Newton method is attractive because it converges to a local minimizzer rapidly from any sufficiently good initial guess. However, it may be expensive to solve the conic Newton equation at each iterate. In this...A conic Newton method is attractive because it converges to a local minimizzer rapidly from any sufficiently good initial guess. However, it may be expensive to solve the conic Newton equation at each iterate. In this paper we consider an inexact conic Newton method, which solves the couic Newton equation oldy approximately and in sonm unspecified manner. Furthermore, we show that such method is locally convergent and characterizes the order of convergence in terms of the rate of convergence of the relative residuals.展开更多
This paper proposes an inexact Newton method via the Lanczos decomposed technique for solving the box-constrained nonlinear systems. An iterative direction is obtained by solving an affine scaling quadratic model with...This paper proposes an inexact Newton method via the Lanczos decomposed technique for solving the box-constrained nonlinear systems. An iterative direction is obtained by solving an affine scaling quadratic model with the Lanczos decomposed technique. By using the interior backtracking line search technique, an acceptable trial step length is found along this direction. The global convergence and the fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the results of the numerical experiments show the effectiveness of the pro- posed algorithm.展开更多
The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting.We employ weak Lipschitz and center-weak Lipschitz conditions to p...The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting.We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis.The obtained results compare favorably with earlier ones such as[7,13,14,18,19].A numerical example is also provided.展开更多
In this study, an interval probability-based inexact two-stage stochastic (IP-ITSP) model is developed for environmental pollutants control and greenhouse gas (GHG) emissions reduction management in regional energy sy...In this study, an interval probability-based inexact two-stage stochastic (IP-ITSP) model is developed for environmental pollutants control and greenhouse gas (GHG) emissions reduction management in regional energy system under uncertainties. In the IP-ITSP model, methods of interval probability, interval-parameter programming (IPP) and two-stage stochastic programming (TSP) are introduced into an integer programming framework;the developed model can tackle uncertainties described in terms of interval values and interval probability distributions. The developed model is applied to a case of planning GHG -emission mitigation in a regional electricity system, demonstrating that IP-ITSP is applicable to reflecting complexities of multi-uncertainty, and capable of addressing the problem of GHG-emission reduction. 4 scenarios corresponding to different GHG -emission mitigation levels are examined;the results indicates that the model could help decision makers identify desired GHG mitigation policies under various economic costs and environmental requirements.展开更多
Bilevel programming problems are a class of optimization problems with hierarchical structure where one of the con-straints is also an optimization problem. Inexact restoration methods were introduced for solving nonl...Bilevel programming problems are a class of optimization problems with hierarchical structure where one of the con-straints is also an optimization problem. Inexact restoration methods were introduced for solving nonlinear programming problems a few years ago. They generate a sequence of, generally, infeasible iterates with intermediate iterations that consist of inexactly restored points. In this paper we present a software environment for solving bilevel program-ming problems using an inexact restoration technique without replacing the lower level problem by its KKT optimality conditions. With this strategy we maintain the minimization structure of the lower level problem and avoid spurious solutions. The environment is a user-friendly set of Fortran 90 modules which is easily and highly configurable. It is prepared to use two well-tested minimization solvers and different formulations in one of the minimization subproblems. We validate our implementation using a set of test problems from the literature, comparing different formulations and the use of the minimization solvers.展开更多
In this paper, an innovative Genetic Algorithms (GA)-based inexact non-linear programming (GAINLP) problem solving approach has been proposed for solving non-linear programming optimization problems with inexact infor...In this paper, an innovative Genetic Algorithms (GA)-based inexact non-linear programming (GAINLP) problem solving approach has been proposed for solving non-linear programming optimization problems with inexact information (inexact non-linear operation programming). GAINLP was developed based on a GA-based inexact quadratic solving method. The Genetic Algorithm Solver of the Global Optimization Toolbox (GASGOT) developed by MATLABTM was adopted as the implementation environment of this study. GAINLP was applied to a municipality solid waste management case. The results from different scenarios indicated that the proposed GA-based heuristic optimization approach was able to generate a solution for a complicated nonlinear problem, which also involved uncertainty.展开更多
The image reconstruction of electrical impedance tomography(EIT)is a nonlinear and ill-posed inverse problem and the imaging results are easily affected by measurement noise,which needs to be solved by using regulariz...The image reconstruction of electrical impedance tomography(EIT)is a nonlinear and ill-posed inverse problem and the imaging results are easily affected by measurement noise,which needs to be solved by using regularization methods.The iterative regularization method has become a focus of the research due to its ease of implementation.To deal with the ill-posed and ill-conditional problems in image reconstruction,the inexact Newton-Landweber iterative method is considered and the Nesterov’s acceleration strategy is introduced.One Nesterov-type accelerated version of the inexact Newton-Landweber iteration is presented to determine the conductivity distributions inside an object from electrical measurements made on the surface.In order to further optimize the acceleration,both the steepest descent step-length and the minimal error step-length are exploited during the iterative image reconstruction process.Landweber iteration and its accelerated version are also implemented for comparison.All algorithms are terminated by the discrepancy principle.Finally,the performance is tested by reporting numerical simulations to verify the remarkable acceleration efficiency of the proposed method.展开更多
The problem of nonconvex and nonsmooth optimization(NNO)has been extensively studied in the machine learning community,leading to the development of numerous fast and convergent numerical algorithms.Existing algorithm...The problem of nonconvex and nonsmooth optimization(NNO)has been extensively studied in the machine learning community,leading to the development of numerous fast and convergent numerical algorithms.Existing algorithms typically employ unified iteration schemes and require explicit solutions to subproblems for ensuring convergence.However,these inflexible iteration schemes overlook task-specific details and may encounter difficulties in providing explicit solutions to subproblems.In contrast,there is evidence suggesting that practical applications can benefit from approximately solving subproblems;however,many existing works fail to establish the theoretical validity of such approximations.In this paper,the authors propose a hybrid inexact proximal alternating method(hiPAM),which addresses a general NNO problem with coupled terms while overcoming all aforementioned challenges.The proposed hiPAM algorithm offers a flexible yet highly efficient approach by seamlessly integrating any efficient methods for approximate subproblem solving that cater to specificities.Additionally,the authors have devised a simple yet implementable stopping criterion that generates a Cauchy sequence and ultimately converges to a critical point of the original NNO problem.The proposed numerical experiments using both simulated and real data have demonstrated that hiPAM represents an exceedingly efficient and robust approach to NNO problems.展开更多
In this paper,we consider the problem of computing the smallest enclosing ball(SEB)of a set of m balls in Rn,where the product mn is large.We first approximate the non-differentiable SEB problem by its log-exponentia...In this paper,we consider the problem of computing the smallest enclosing ball(SEB)of a set of m balls in Rn,where the product mn is large.We first approximate the non-differentiable SEB problem by its log-exponential aggregation function and then propose a computationally efficient inexact Newton-CG algorithm for the smoothing approximation problem by exploiting its special(approximate)sparsity structure.The key difference between the proposed inexact Newton-CG algorithm and the classical Newton-CG algorithm is that the gradient and the Hessian-vector product are inexactly computed in the proposed algorithm,which makes it capable of solving the large-scale SEB problem.We give an adaptive criterion of inexactly computing the gradient/Hessian and establish global convergence of the proposed algorithm.We illustrate the efficiency of the proposed algorithm by using the classical Newton-CG algorithm as well as the algorithm from Zhou et al.(Comput Optim Appl 30:147–160,2005)as benchmarks.展开更多
Inexact Newton methods are constructed by combining Newton＇s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems f...Inexact Newton methods are constructed by combining Newton＇s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems for the inexact Newton methods. When these two theorems are specified to Newton＇s method, we obtain a different Newton-Kantorovich theorem about Newton＇s method. When the iterative method for solving the Newton equations is specified to be the splitting method, we get two estimates about the iteration steps for the special inexact Newton methods.展开更多
We propose an inexact affine scaling Levenberg-Marquardt method for solving bound-constrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition...We propose an inexact affine scaling Levenberg-Marquardt method for solving bound-constrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition. The affine scaling Levenberg-Marquardt equation is based on a minimization of the squared Euclidean norm of linearized model adding a quadratic affine scaling matrix to find a solution which belongs to the bounded constraints on variable. The global convergence and the superlinear convergence rate are proved.Numerical results show that the new algorithm is efficient.展开更多
We first propose a new class of smoothing functions for the non- linear complementarity function which contains the well-known Chen-Harker- Kanzow-Smale smoothing function and Huang-Han-Chen smoothing function as spec...We first propose a new class of smoothing functions for the non- linear complementarity function which contains the well-known Chen-Harker- Kanzow-Smale smoothing function and Huang-Han-Chen smoothing function as special cases, and then present a smoothing inexact Newton algorithm for the P0 nonlinear complementarity problem. The global convergence and local superlinear convergence are established. Preliminary numerical results indicate the feasibility and efficiency of the algorithm.展开更多
For the Hermitian inexact Rayleigh quotient iteration （RQI）, we consider the local convergence of the inexact RQI with the Lanczos method for the linear systems involved. Some attractive properties are derived for t...For the Hermitian inexact Rayleigh quotient iteration （RQI）, we consider the local convergence of the inexact RQI with the Lanczos method for the linear systems involved. Some attractive properties are derived for the residual, whose norm is ξk, of the linear system obtained by the Lanczos method at outer iteration k ＋ 1. Based on them, we make a refined analysis and establish new local convergence results. It is proved that （i） the inexact RQI with Lanezos converges quadratically provided that ξk ≤ξ with a constant ξ≥） 1 and （ii） the method converges linearly provided that ξk is bounded by some multiple of 1/‖τk‖ with ‖τk‖ the residual norm of the approximate eigenpair at outer iteration k. The results are fundamentally different from the existing ones that always require ξk 〈 1, and they have implications on effective implementations of the method. Based on the new theory, we can design practical criteria to control ξk to achieve quadratic convergence and implement the method more effectively than ever before. Numerical experiments confirm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.展开更多
Nonnegative tensor decomposition has become increasingly important for multiway data analysis in recent years. The alternating proximal gradient(APG) is a popular optimization method for nonnegative tensor decompositi...Nonnegative tensor decomposition has become increasingly important for multiway data analysis in recent years. The alternating proximal gradient(APG) is a popular optimization method for nonnegative tensor decomposition in the block coordinate descent framework. In this study, we propose an inexact version of the APG algorithm for nonnegative CANDECOMP/PARAFAC decomposition, wherein each factor matrix is updated by only finite inner iterations. We also propose a parameter warm-start method that can avoid the frequent parameter resetting of conventional APG methods and improve convergence performance.By experimental tests, we find that when the number of inner iterations is limited to around 10 to 20, the convergence speed is accelerated significantly without losing its low relative error. We evaluate our method on both synthetic and real-world tensors.The results demonstrate that the proposed inexact APG algorithm exhibits outstanding performance on both convergence speed and computational precision compared with existing popular algorithms.展开更多
In this paper, a S-CR method with inexact solvers on the subdomains is presented, and then its convergence property is proved under very general conditions. This result is important because it guarantees the effective...In this paper, a S-CR method with inexact solvers on the subdomains is presented, and then its convergence property is proved under very general conditions. This result is important because it guarantees the effectiveness of the Schwarz alternating method when executed on message-passing distributed memory multiprocessor system.展开更多
The proximal alternating linearized minimization(PALM)method suits well for solving blockstructured optimization problems,which are ubiquitous in real applications.In the cases where subproblems do not have closed-for...The proximal alternating linearized minimization(PALM)method suits well for solving blockstructured optimization problems,which are ubiquitous in real applications.In the cases where subproblems do not have closed-form solutions,e.g.,due to complex constraints,infeasible subsolvers are indispensable,giving rise to an infeasible inexact PALM(PALM-I).Numerous efforts have been devoted to analyzing the feasible PALM,while little attention has been paid to the PALM-I.The usage of the PALM-I thus lacks a theoretical guarantee.The essential difficulty of analysis consists in the objective value nonmonotonicity induced by the infeasibility.We study in the present work the convergence properties of the PALM-I.In particular,we construct a surrogate sequence to surmount the nonmonotonicity issue and devise an implementable inexact criterion.Based upon these,we manage to establish the stationarity of any accumulation point,and moreover,show the iterate convergence and the asymptotic convergence rates under the assumption of the Lojasiewicz property.The prominent advantages of the PALM-I on CPU time are illustrated via numerical experiments on problems arising from quantum physics and 3-dimensional anisotropic frictional contact.展开更多
基金Supported by National Natural Science Foundation of China (No.51275348)College Students Innovation and Entrepreneurship Training Program of Tianjin University (No.201210056339)
文摘In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.
文摘A trust-region algorithm is presented for a nonlinear optimization problem of equality-constraints. The characterization of the algorithm is using inexact gradient information. Global convergence results are demonstrated where the gradient values are obeyed a simple relative error condition.
文摘A conic Newton method is attractive because it converges to a local minimizzer rapidly from any sufficiently good initial guess. However, it may be expensive to solve the conic Newton equation at each iterate. In this paper we consider an inexact conic Newton method, which solves the couic Newton equation oldy approximately and in sonm unspecified manner. Furthermore, we show that such method is locally convergent and characterizes the order of convergence in terms of the rate of convergence of the relative residuals.
基金Project supported by the National Natural Science Foundation of China (No. 10871130)the Ph. D.Programs Foundation of Ministry of Education of China (No. 20093127110005)the Shanghai Leading Academic Discipline Project (No. T0401)
文摘This paper proposes an inexact Newton method via the Lanczos decomposed technique for solving the box-constrained nonlinear systems. An iterative direction is obtained by solving an affine scaling quadratic model with the Lanczos decomposed technique. By using the interior backtracking line search technique, an acceptable trial step length is found along this direction. The global convergence and the fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the results of the numerical experiments show the effectiveness of the pro- posed algorithm.
文摘The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting.We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis.The obtained results compare favorably with earlier ones such as[7,13,14,18,19].A numerical example is also provided.
文摘In this study, an interval probability-based inexact two-stage stochastic (IP-ITSP) model is developed for environmental pollutants control and greenhouse gas (GHG) emissions reduction management in regional energy system under uncertainties. In the IP-ITSP model, methods of interval probability, interval-parameter programming (IPP) and two-stage stochastic programming (TSP) are introduced into an integer programming framework;the developed model can tackle uncertainties described in terms of interval values and interval probability distributions. The developed model is applied to a case of planning GHG -emission mitigation in a regional electricity system, demonstrating that IP-ITSP is applicable to reflecting complexities of multi-uncertainty, and capable of addressing the problem of GHG-emission reduction. 4 scenarios corresponding to different GHG -emission mitigation levels are examined;the results indicates that the model could help decision makers identify desired GHG mitigation policies under various economic costs and environmental requirements.
文摘Bilevel programming problems are a class of optimization problems with hierarchical structure where one of the con-straints is also an optimization problem. Inexact restoration methods were introduced for solving nonlinear programming problems a few years ago. They generate a sequence of, generally, infeasible iterates with intermediate iterations that consist of inexactly restored points. In this paper we present a software environment for solving bilevel program-ming problems using an inexact restoration technique without replacing the lower level problem by its KKT optimality conditions. With this strategy we maintain the minimization structure of the lower level problem and avoid spurious solutions. The environment is a user-friendly set of Fortran 90 modules which is easily and highly configurable. It is prepared to use two well-tested minimization solvers and different formulations in one of the minimization subproblems. We validate our implementation using a set of test problems from the literature, comparing different formulations and the use of the minimization solvers.
文摘In this paper, an innovative Genetic Algorithms (GA)-based inexact non-linear programming (GAINLP) problem solving approach has been proposed for solving non-linear programming optimization problems with inexact information (inexact non-linear operation programming). GAINLP was developed based on a GA-based inexact quadratic solving method. The Genetic Algorithm Solver of the Global Optimization Toolbox (GASGOT) developed by MATLABTM was adopted as the implementation environment of this study. GAINLP was applied to a municipality solid waste management case. The results from different scenarios indicated that the proposed GA-based heuristic optimization approach was able to generate a solution for a complicated nonlinear problem, which also involved uncertainty.
基金National Natural Science Foundation of China(12101204,12261021)Heilongjiang Provincial Natural Science Foundation of China(LH2023A018)Modern Numerical Method Course for Research Program on Teaching Reform of Degree and Postgraduate Education of Heilongjiang University(2024)。
文摘The image reconstruction of electrical impedance tomography(EIT)is a nonlinear and ill-posed inverse problem and the imaging results are easily affected by measurement noise,which needs to be solved by using regularization methods.The iterative regularization method has become a focus of the research due to its ease of implementation.To deal with the ill-posed and ill-conditional problems in image reconstruction,the inexact Newton-Landweber iterative method is considered and the Nesterov’s acceleration strategy is introduced.One Nesterov-type accelerated version of the inexact Newton-Landweber iteration is presented to determine the conductivity distributions inside an object from electrical measurements made on the surface.In order to further optimize the acceleration,both the steepest descent step-length and the minimal error step-length are exploited during the iterative image reconstruction process.Landweber iteration and its accelerated version are also implemented for comparison.All algorithms are terminated by the discrepancy principle.Finally,the performance is tested by reporting numerical simulations to verify the remarkable acceleration efficiency of the proposed method.
基金supported by the National Key R&D Program of China under Grant No.2023YFA1011303the National Natural Science Foundation of China under Grant Nos.61806057 and 12301479+1 种基金the China Postdoctoral Science Foundation under Grant No.2018M632018the Natural Science Foundation of Liaoning Province under Grant No.2023-MS-126。
文摘The problem of nonconvex and nonsmooth optimization(NNO)has been extensively studied in the machine learning community,leading to the development of numerous fast and convergent numerical algorithms.Existing algorithms typically employ unified iteration schemes and require explicit solutions to subproblems for ensuring convergence.However,these inflexible iteration schemes overlook task-specific details and may encounter difficulties in providing explicit solutions to subproblems.In contrast,there is evidence suggesting that practical applications can benefit from approximately solving subproblems;however,many existing works fail to establish the theoretical validity of such approximations.In this paper,the authors propose a hybrid inexact proximal alternating method(hiPAM),which addresses a general NNO problem with coupled terms while overcoming all aforementioned challenges.The proposed hiPAM algorithm offers a flexible yet highly efficient approach by seamlessly integrating any efficient methods for approximate subproblem solving that cater to specificities.Additionally,the authors have devised a simple yet implementable stopping criterion that generates a Cauchy sequence and ultimately converges to a critical point of the original NNO problem.The proposed numerical experiments using both simulated and real data have demonstrated that hiPAM represents an exceedingly efficient and robust approach to NNO problems.
基金the National Natural Science Foundation of China(Nos.11331012 and 11301516).
文摘In this paper,we consider the problem of computing the smallest enclosing ball(SEB)of a set of m balls in Rn,where the product mn is large.We first approximate the non-differentiable SEB problem by its log-exponential aggregation function and then propose a computationally efficient inexact Newton-CG algorithm for the smoothing approximation problem by exploiting its special(approximate)sparsity structure.The key difference between the proposed inexact Newton-CG algorithm and the classical Newton-CG algorithm is that the gradient and the Hessian-vector product are inexactly computed in the proposed algorithm,which makes it capable of solving the large-scale SEB problem.We give an adaptive criterion of inexactly computing the gradient/Hessian and establish global convergence of the proposed algorithm.We illustrate the efficiency of the proposed algorithm by using the classical Newton-CG algorithm as well as the algorithm from Zhou et al.(Comput Optim Appl 30:147–160,2005)as benchmarks.
基金Supported by State Key Laboratory of Scientific/Engineering Computing,Chinese Academy of Sciencesthe National Natural Science Foundation of China (10571059,10571060).
文摘Inexact Newton methods are constructed by combining Newton＇s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems for the inexact Newton methods. When these two theorems are specified to Newton＇s method, we obtain a different Newton-Kantorovich theorem about Newton＇s method. When the iterative method for solving the Newton equations is specified to be the splitting method, we get two estimates about the iteration steps for the special inexact Newton methods.
基金Supported by National Natural Science Foundation of China(No.11571074)Scientific Research Fund of Hunan Provincial Education Department(No.18A351,17C0393)Natural Science Foundation of Hunan Province(No.2019JJ50105)
文摘We propose an inexact affine scaling Levenberg-Marquardt method for solving bound-constrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition. The affine scaling Levenberg-Marquardt equation is based on a minimization of the squared Euclidean norm of linearized model adding a quadratic affine scaling matrix to find a solution which belongs to the bounded constraints on variable. The global convergence and the superlinear convergence rate are proved.Numerical results show that the new algorithm is efficient.
文摘We first propose a new class of smoothing functions for the non- linear complementarity function which contains the well-known Chen-Harker- Kanzow-Smale smoothing function and Huang-Han-Chen smoothing function as special cases, and then present a smoothing inexact Newton algorithm for the P0 nonlinear complementarity problem. The global convergence and local superlinear convergence are established. Preliminary numerical results indicate the feasibility and efficiency of the algorithm.
基金supported by National Basic Research Program of China(Grant No.2011CB302400)National Natural Science Foundation of China(Grant No.11071140)
文摘For the Hermitian inexact Rayleigh quotient iteration （RQI）, we consider the local convergence of the inexact RQI with the Lanczos method for the linear systems involved. Some attractive properties are derived for the residual, whose norm is ξk, of the linear system obtained by the Lanczos method at outer iteration k ＋ 1. Based on them, we make a refined analysis and establish new local convergence results. It is proved that （i） the inexact RQI with Lanezos converges quadratically provided that ξk ≤ξ with a constant ξ≥） 1 and （ii） the method converges linearly provided that ξk is bounded by some multiple of 1/‖τk‖ with ‖τk‖ the residual norm of the approximate eigenpair at outer iteration k. The results are fundamentally different from the existing ones that always require ξk 〈 1, and they have implications on effective implementations of the method. Based on the new theory, we can design practical criteria to control ξk to achieve quadratic convergence and implement the method more effectively than ever before. Numerical experiments confirm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.
基金This work was supported by the National Natural Science Foundation of China(Grant No.91748105)the National Foundation in China(Grant Nos.JCKY2019110B009 and 2020-JCJQ-JJ-252)+1 种基金the Fundamental Research Funds for the Central Universities(Grant Nos.DUT20LAB303 and DUT20LAB308)in Dalian University of Technology in Chinathe scholarship from China Scholarship Council(Grant No.201600090043)。
文摘Nonnegative tensor decomposition has become increasingly important for multiway data analysis in recent years. The alternating proximal gradient(APG) is a popular optimization method for nonnegative tensor decomposition in the block coordinate descent framework. In this study, we propose an inexact version of the APG algorithm for nonnegative CANDECOMP/PARAFAC decomposition, wherein each factor matrix is updated by only finite inner iterations. We also propose a parameter warm-start method that can avoid the frequent parameter resetting of conventional APG methods and improve convergence performance.By experimental tests, we find that when the number of inner iterations is limited to around 10 to 20, the convergence speed is accelerated significantly without losing its low relative error. We evaluate our method on both synthetic and real-world tensors.The results demonstrate that the proposed inexact APG algorithm exhibits outstanding performance on both convergence speed and computational precision compared with existing popular algorithms.
文摘In this paper, a S-CR method with inexact solvers on the subdomains is presented, and then its convergence property is proved under very general conditions. This result is important because it guarantees the effectiveness of the Schwarz alternating method when executed on message-passing distributed memory multiprocessor system.
基金supported by National Natural Science Foundation of China(Grant Nos.12125108,11971466,11991021,11991020,12021001 and 12288201)Key Research Program of Frontier Sciences,Chinese Academy of Sciences(Grant No.ZDBS-LY-7022)CAS(the Chinese Academy of Sciences)AMSS(Academy of Mathematics and Systems Science)-PolyU(The Hong Kong Polytechnic University)Joint Laboratory of Applied Mathematics.
文摘The proximal alternating linearized minimization(PALM)method suits well for solving blockstructured optimization problems,which are ubiquitous in real applications.In the cases where subproblems do not have closed-form solutions,e.g.,due to complex constraints,infeasible subsolvers are indispensable,giving rise to an infeasible inexact PALM(PALM-I).Numerous efforts have been devoted to analyzing the feasible PALM,while little attention has been paid to the PALM-I.The usage of the PALM-I thus lacks a theoretical guarantee.The essential difficulty of analysis consists in the objective value nonmonotonicity induced by the infeasibility.We study in the present work the convergence properties of the PALM-I.In particular,we construct a surrogate sequence to surmount the nonmonotonicity issue and devise an implementable inexact criterion.Based upon these,we manage to establish the stationarity of any accumulation point,and moreover,show the iterate convergence and the asymptotic convergence rates under the assumption of the Lojasiewicz property.The prominent advantages of the PALM-I on CPU time are illustrated via numerical experiments on problems arising from quantum physics and 3-dimensional anisotropic frictional contact.
