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
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 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
Classical quasi-Newton methods are widely used to solve nonlinear problems in which the first-order information is exact.In some practical problems,we can only obtain approximate values of the objective function and i...Classical quasi-Newton methods are widely used to solve nonlinear problems in which the first-order information is exact.In some practical problems,we can only obtain approximate values of the objective function and its gradient.It is necessary to design optimization algorithms that can utilize inexact first-order information.In this paper,we propose an adaptive regularized quasi-Newton method to solve such problems.Under some mild conditions,we prove the global convergence and establish the convergence rate of the adaptive regularized quasi-Newton method.Detailed implementations of our method,including the subspace technique to reduce the amount of computation,are presented.Encouraging numerical results demonstrate that the adaptive regularized quasi-Newton method is a promising method,which can utilize the inexact first-order information effectively.展开更多
Optimization problem of cardinality constrained mean-variance(CCMV)model for sparse portfolio selection is considered.To overcome the difficulties caused by cardinality constraint,an exact penalty approach is employed...Optimization problem of cardinality constrained mean-variance(CCMV)model for sparse portfolio selection is considered.To overcome the difficulties caused by cardinality constraint,an exact penalty approach is employed,then CCMV problem is transferred into a difference-of-convex-functions(DC)problem.By exploiting the DC structure of the gained problem and the superlinear convergence of semismooth Newton(ssN)method,an inexact proximal DC algorithm with sieving strategy based on a majorized ssN method(siPDCA-mssN)is proposed.For solving the inner problems of siPDCA-mssN from dual,the second-order information is wisely incorporated and an efficient mssN method is employed.The global convergence of the sequence generated by siPDCA-mssN is proved.To solve large-scale CCMV problem,a decomposed siPDCA-mssN(DsiPDCA-mssN)is introduced.To demonstrate the efficiency of proposed algorithms,siPDCA-mssN and DsiPDCA-mssN are compared with the penalty proximal alternating linearized minimization method and the CPLEX(12.9)solver by performing numerical experiments on realword market data and large-scale simulated data.The numerical results demonstrate that siPDCA-mssN and DsiPDCA-mssN outperform the other methods from computation time and optimal value.The out-of-sample experiments results display that the solutions of CCMV model are better than those of other portfolio selection models in terms of Sharp ratio and sparsity.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
随着人工智能对算力需求的激增,数据中心(internet data center,IDC)作为数据处理与存储的机构,其能耗需求远超预期,使用新能源是其可持续发展的需要。然而,可再生能源具有出力不确定性,仅依靠数据中心参与需求响应难以实现消纳,可配置...随着人工智能对算力需求的激增,数据中心(internet data center,IDC)作为数据处理与存储的机构,其能耗需求远超预期,使用新能源是其可持续发展的需要。然而,可再生能源具有出力不确定性,仅依靠数据中心参与需求响应难以实现消纳,可配置储能提高系统灵活性。因此,本工作建立了以规划总成本最优为目标的数据中心与电池储能(battery energy storage,BES)协同规划两阶段鲁棒模型,为防止规划结果过于乐观,引入了储能寿命约束。同时针对在求解所建模型过程中,传统C&CG(column-and-constraint generation)算法存在难以平衡求解速度与精度间关系的问题,本工作提出了一种不精确列和生成约束算法i-C&CG(inexact column-and-constraint generation)进行求解。基于IEEE30节点与IEEE118节点算例系统进行优化解算,仿真结果表明,与仅配置单一储能相比,本工作所提模型储能年等效建设成本下降39785元,数据中心年等效建设成本下降289080元;且本工作所提算法与传统C&CG相比,采用0.18低精度下的i-C&CG,与采用0.16较高精度的C&CG相比较,i-C&CG最多可缩短3632 s的单次迭代求解所需时间,且最终计算结果的相对误差为0.46%,两者收敛间隙与相对最优间隙近似。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
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 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supported by the National Natural Science Foundation of China(Grant No.NSFC-11971118).
文摘Classical quasi-Newton methods are widely used to solve nonlinear problems in which the first-order information is exact.In some practical problems,we can only obtain approximate values of the objective function and its gradient.It is necessary to design optimization algorithms that can utilize inexact first-order information.In this paper,we propose an adaptive regularized quasi-Newton method to solve such problems.Under some mild conditions,we prove the global convergence and establish the convergence rate of the adaptive regularized quasi-Newton method.Detailed implementations of our method,including the subspace technique to reduce the amount of computation,are presented.Encouraging numerical results demonstrate that the adaptive regularized quasi-Newton method is a promising method,which can utilize the inexact first-order information effectively.
基金supported by the National Natural Science Foundation of China(Grant No.11971092)supported by the Fundamental Research Funds for the Central Universities(Grant No.DUT20RC(3)079)。
文摘Optimization problem of cardinality constrained mean-variance(CCMV)model for sparse portfolio selection is considered.To overcome the difficulties caused by cardinality constraint,an exact penalty approach is employed,then CCMV problem is transferred into a difference-of-convex-functions(DC)problem.By exploiting the DC structure of the gained problem and the superlinear convergence of semismooth Newton(ssN)method,an inexact proximal DC algorithm with sieving strategy based on a majorized ssN method(siPDCA-mssN)is proposed.For solving the inner problems of siPDCA-mssN from dual,the second-order information is wisely incorporated and an efficient mssN method is employed.The global convergence of the sequence generated by siPDCA-mssN is proved.To solve large-scale CCMV problem,a decomposed siPDCA-mssN(DsiPDCA-mssN)is introduced.To demonstrate the efficiency of proposed algorithms,siPDCA-mssN and DsiPDCA-mssN are compared with the penalty proximal alternating linearized minimization method and the CPLEX(12.9)solver by performing numerical experiments on realword market data and large-scale simulated data.The numerical results demonstrate that siPDCA-mssN and DsiPDCA-mssN outperform the other methods from computation time and optimal value.The out-of-sample experiments results display that the solutions of CCMV model are better than those of other portfolio selection models in terms of Sharp ratio and sparsity.
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.
文摘随着人工智能对算力需求的激增,数据中心(internet data center,IDC)作为数据处理与存储的机构,其能耗需求远超预期,使用新能源是其可持续发展的需要。然而,可再生能源具有出力不确定性,仅依靠数据中心参与需求响应难以实现消纳,可配置储能提高系统灵活性。因此,本工作建立了以规划总成本最优为目标的数据中心与电池储能(battery energy storage,BES)协同规划两阶段鲁棒模型,为防止规划结果过于乐观,引入了储能寿命约束。同时针对在求解所建模型过程中,传统C&CG(column-and-constraint generation)算法存在难以平衡求解速度与精度间关系的问题,本工作提出了一种不精确列和生成约束算法i-C&CG(inexact column-and-constraint generation)进行求解。基于IEEE30节点与IEEE118节点算例系统进行优化解算,仿真结果表明,与仅配置单一储能相比,本工作所提模型储能年等效建设成本下降39785元,数据中心年等效建设成本下降289080元;且本工作所提算法与传统C&CG相比,采用0.18低精度下的i-C&CG,与采用0.16较高精度的C&CG相比较,i-C&CG最多可缩短3632 s的单次迭代求解所需时间,且最终计算结果的相对误差为0.46%,两者收敛间隙与相对最优间隙近似。
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.
