In this paper,we propose a new full-Newton step feasible interior-point algorithm for the special weighted linear complementarity problems.The proposed algorithm employs the technique of algebraic equivalent transform...In this paper,we propose a new full-Newton step feasible interior-point algorithm for the special weighted linear complementarity problems.The proposed algorithm employs the technique of algebraic equivalent transformation to derive the search direction.It is shown that the proximity measure reduces quadratically at each iteration.Moreover,the iteration bound of the algorithm is as good as the best-known polynomial complexity for these types of problems.Furthermore,numerical results are presented to show the efficiency of the proposed algorithm.展开更多
A family of neural networks is proposed to solve linear complementarity problems(LCP).The neural networks are constructed from the novel equivalent model of LCP,which is reformulated by utilizing the modulus and smoot...A family of neural networks is proposed to solve linear complementarity problems(LCP).The neural networks are constructed from the novel equivalent model of LCP,which is reformulated by utilizing the modulus and smoothing technologies.Some important properties of the proposed novel equivalent model are summarized.In addition,the stability properties of the proposed steepest descent-based neural networks for LCP are analyzed.In order to illustrate the theoretical results,we provide some numerical simulations and compare the proposed neural networks with existing neural networks based on the NCP-functions.Numerical results indicate that the performance of the proposed neural networks is effective and robust.展开更多
In this paper,a new full-Newton step primal-dual interior-point algorithm for solving the special weighted linear complementarity problem is designed and analyzed.The algorithm employs a kernel function with a linear ...In this paper,a new full-Newton step primal-dual interior-point algorithm for solving the special weighted linear complementarity problem is designed and analyzed.The algorithm employs a kernel function with a linear growth term to derive the search direction,and by introducing new technical results and selecting suitable parameters,we prove that the iteration bound of the algorithm is as good as best-known polynomial complexity of interior-point methods.Furthermore,numerical results illustrate the efficiency of the proposed method.展开更多
In this paper,a two-step iteration method is established which can be viewed as a generalization of the existing modulus-based methods for vertical linear complementarity problems given by He and Vong(Appl.Math.Lett.1...In this paper,a two-step iteration method is established which can be viewed as a generalization of the existing modulus-based methods for vertical linear complementarity problems given by He and Vong(Appl.Math.Lett.134:108344,2022).The convergence analysis of the proposed method is established,which can improve the existing results.Numerical examples show that the proposed method is efficient with the two-step technique.展开更多
A partition reduction method is used to obtain new upper bounds for the inverses of H-matrices and S-strictly diagonally dominant(S-SDD)matrices.The estimates are expressed via the determinants of third order matrices...A partition reduction method is used to obtain new upper bounds for the inverses of H-matrices and S-strictly diagonally dominant(S-SDD)matrices.The estimates are expressed via the determinants of third order matrices.Numerical experiments with various random matrices show that they are stable and better than the estimates presented in literatures.We use these upper bounds to improve known error estimates for linear complementarity problems with H-matrices and S-SDD matrices.展开更多
This paper addresses the generalized linear complementarity problem （GLCP） over a polyhedral cone. To solve the problem, we first equivalently convert the problem into an affine variational inequalities problem over...This paper addresses the generalized linear complementarity problem （GLCP） over a polyhedral cone. To solve the problem, we first equivalently convert the problem into an affine variational inequalities problem over a closed polyhedral cone, and then propose a new type of method to solve the GLCP based on the error bound estimation. The global and R-linear convergence rate is established. The numerical experiments show the efficiency of the method.展开更多
In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient proje...In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient projection method is given for solving the stochastic generalized linear complementarity problems. The global convergence of the conjugate gradient projection method is proved and the related numerical results are also reported.展开更多
Feasible-interior-point algorithms start from a strictly feasible interior point,but infeassible-interior-point algorithms just need to start from an arbitrary positive point,we give a potential reduction algorithm fr...Feasible-interior-point algorithms start from a strictly feasible interior point,but infeassible-interior-point algorithms just need to start from an arbitrary positive point,we give a potential reduction algorithm from an infeasible-starting-point for a class of non-monotone linear complementarity problem.Its polynomial complexity is analyzed.After finite iterations the algorithm produces an approximate solution of the problem or shows that there is no feasible optimal solution in a large region.展开更多
A one_step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so_called aggregation function. The proposed algorithm has the following good features: (ⅰ) It solve...A one_step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so_called aggregation function. The proposed algorithm has the following good features: (ⅰ) It solves only one linear system of equations and does only one line search at each iteration; (ⅱ) It is well_defined for the vertical linear complementarity problem with vertical block P 0 matrix and any accumulation point of iteration sequence is its solution.Moreover, the iteration sequence is bounded for the vertical linear complementarity problem with vertical block P 0+R 0 matrix; (ⅲ) It has both global linear and local quadratic convergence without strict complementarity. Many existing smoothing Newton methods do not have the property (ⅲ).展开更多
This paper proposes an infeasible interior-point algorithm for linear complementarity problem with full-Newton steps.The main iteration consists of a feasibility step and several centrality steps.No more than O（n log...This paper proposes an infeasible interior-point algorithm for linear complementarity problem with full-Newton steps.The main iteration consists of a feasibility step and several centrality steps.No more than O（n log（n /ε））iterations are required for getting ε-solution of the problem at hand,which coincides with the best-known bound for infeasible interior-point algorithms.展开更多
It has been shown in various papers that most interior-point algorithms for linear optimization and their analysis can be generalized to P_＊（κ） linear complementarity problems.This paper presents an extension of t...It has been shown in various papers that most interior-point algorithms for linear optimization and their analysis can be generalized to P_＊（κ） linear complementarity problems.This paper presents an extension of the recent variant of Mehrotra＇s second order algorithm for linear optimijation.It is shown that the iteration-complexity bound of the algorithm is O（4κ ＋ 3）√14κ ＋ 5 nlog（x0）Ts0/ε,which is similar to that of the corresponding algorithm for linear optimization.展开更多
A polynomial interior-point algorithm is presented for monotone linear complementarity problem （MLCP） based on a class of kernel functions with the general barrier term, which are called general kernel functions. Un...A polynomial interior-point algorithm is presented for monotone linear complementarity problem （MLCP） based on a class of kernel functions with the general barrier term, which are called general kernel functions. Under the mild conditions for the barrier term, the complexity bound of algorithm in terms of such kernel function and its derivatives is obtained. The approach is actually an extension of the existing work which only used the specific kernel functions for the MLCP.展开更多
A modified sequential linear programming algorithm is presented, whose subproblem is always solvable, for the extended linear complementarity problem (XLCP), the global convergence of the algorithm under assumption of...A modified sequential linear programming algorithm is presented, whose subproblem is always solvable, for the extended linear complementarity problem (XLCP), the global convergence of the algorithm under assumption of X-row sufficiency or X-colunm monotonicity is proved. As a result, a sufficient condition for existence and boundedness of solution to the XLCP are obtained.展开更多
Mehrotra-type predictor-corrector algorithm, as one of most efficient interior point methods, has become the backbones of most optimization packages. Salahi et al. proposed a cut strategy based algorithm for linear op...Mehrotra-type predictor-corrector algorithm, as one of most efficient interior point methods, has become the backbones of most optimization packages. Salahi et al. proposed a cut strategy based algorithm for linear optimization that enjoyed polynomial complexity and maintained its efficiency in practice. We extend their algorithm to P. （~） linear complementar- ity problems. The way of choosing corrector direction for our algorithm is different from theirs： The new algorithm has been proved to have an O（（1 ＋ 4k）（17 ＋ 19k）√1＋2kn 3/2 log（x0）Ts0/ε） worst case iteration complexity bound. An numerical experiment verifies the feasibility of the new algorithm.展开更多
The extended linear complementarity problem(denoted by ELCP) can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed CHKS smoothing function, the ELCP is approximated by ...The extended linear complementarity problem(denoted by ELCP) can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed CHKS smoothing function, the ELCP is approximated by a family of parameterized smooth equations. A one-step smoothing Newton method is designed for solving the ELCP. The proposed algorithm is proved to be globally convergent under suitable assumptions.展开更多
In this paper,by means of constructing the linear complementarity problems into the corresponding absolute value equation,we raise an iteration method,called as the nonlinear lopsided HSS-like modulus-based matrix spl...In this paper,by means of constructing the linear complementarity problems into the corresponding absolute value equation,we raise an iteration method,called as the nonlinear lopsided HSS-like modulus-based matrix splitting iteration method,for solving the linear complementarity problems whose coefficient matrix in R^(n×n)is large sparse and positive definite.From the convergence analysis,it is appreciable to see that the proposed method will converge to its accurate solution under appropriate conditions.Numerical examples demonstrate that the presented method precede to other methods in practical implementation.展开更多
This paper considers semidefinite relaxation for linear and nonlinear complementarity problems.For some particular copositive matrices and tensors,the existence of a solution for the corresponding complementarity prob...This paper considers semidefinite relaxation for linear and nonlinear complementarity problems.For some particular copositive matrices and tensors,the existence of a solution for the corresponding complementarity problems is studied.Under a general assumption,we show that if the solution set of a complementarity problem is nonempty,then we can get a solution by the semidefinite relaxation method;while if it does not have a solution,we can obtain a certificate for the infeasibility.Some numerical examples are given.展开更多
In this paper,we discuss the perturbation analysis of the extended vertical linear complementarity problem(EVLCP).Under the assumption of the row W-property,we derive several absolute and relative perturbation bounds ...In this paper,we discuss the perturbation analysis of the extended vertical linear complementarity problem(EVLCP).Under the assumption of the row W-property,we derive several absolute and relative perturbation bounds of EVLCP,which extend some existing results.Several numerical examples are given to show the proposed bounds.展开更多
Recently, we have proposed an iterative projection and contraction (PC) method for a class of linear complementarity problems (LCP)([4]). The method was showed to be globally convergent, but no statement could be made...Recently, we have proposed an iterative projection and contraction (PC) method for a class of linear complementarity problems (LCP)([4]). The method was showed to be globally convergent, but no statement could be made about the rate of convergence. In this paper, we develop a modified globally linearly convergent PC method for linear complementarity problems. Both the method and the convergence proofs are very simple. The method can also be used to solve some linear variational inequalities. Several computational experiments are presented to indicate that the method is surprising good for solving some known difficult problems.展开更多
Asynchronous parallel multisplitting relaxation methods for solving large sparse linear complementarity problems are presented, and their convergence is proved when the system matrices are H-matrices having positive d...Asynchronous parallel multisplitting relaxation methods for solving large sparse linear complementarity problems are presented, and their convergence is proved when the system matrices are H-matrices having positive diagonal elements. Moreover, block and multi-parameter variants of the new methods, together with their convergence properties, are investigated in detail. Numerical results show that these new methods can achieve high parallel efficiency for solving the large sparse linear complementarity problems on multiprocessor systems.展开更多
基金Supported by the Optimisation Theory and Algorithm Research Team(Grant No.23kytdzd004)University Science Research Project of Anhui Province(Grant No.2024AH050631)the General Programs for Young Teacher Cultivation of Educational Commission of Anhui Province(Grant No.YQYB2023090).
文摘In this paper,we propose a new full-Newton step feasible interior-point algorithm for the special weighted linear complementarity problems.The proposed algorithm employs the technique of algebraic equivalent transformation to derive the search direction.It is shown that the proximity measure reduces quadratically at each iteration.Moreover,the iteration bound of the algorithm is as good as the best-known polynomial complexity for these types of problems.Furthermore,numerical results are presented to show the efficiency of the proposed algorithm.
基金Supported by the National Natural Science Foundation of China(12371378,41725017,11901098).
文摘A family of neural networks is proposed to solve linear complementarity problems(LCP).The neural networks are constructed from the novel equivalent model of LCP,which is reformulated by utilizing the modulus and smoothing technologies.Some important properties of the proposed novel equivalent model are summarized.In addition,the stability properties of the proposed steepest descent-based neural networks for LCP are analyzed.In order to illustrate the theoretical results,we provide some numerical simulations and compare the proposed neural networks with existing neural networks based on the NCP-functions.Numerical results indicate that the performance of the proposed neural networks is effective and robust.
基金Supported by University Science Research Project of Anhui Province(2023AH052921)Outstanding Youth Talent Project of Anhui Province(gxyq2021254)。
文摘In this paper,a new full-Newton step primal-dual interior-point algorithm for solving the special weighted linear complementarity problem is designed and analyzed.The algorithm employs a kernel function with a linear growth term to derive the search direction,and by introducing new technical results and selecting suitable parameters,we prove that the iteration bound of the algorithm is as good as best-known polynomial complexity of interior-point methods.Furthermore,numerical results illustrate the efficiency of the proposed method.
基金supported by the Scientific Computing Research Innovation Team of Guangdong Province(no.2021KCXTD052)the Science and Technology Development Fund,Macao SAR(no.0096/2022/A,0151/2022/A)+3 种基金University of Macao(no.MYRG2020-00035-FST,MYRG2022-00076-FST)the Guangdong Key Construction Discipline Research Capacity Enhancement Project(no.2022ZDJS049)Technology Planning Project of Shaoguan(no.210716094530390)the ScienceFoundation of Shaoguan University(no.SZ2020KJ01).
文摘In this paper,a two-step iteration method is established which can be viewed as a generalization of the existing modulus-based methods for vertical linear complementarity problems given by He and Vong(Appl.Math.Lett.134:108344,2022).The convergence analysis of the proposed method is established,which can improve the existing results.Numerical examples show that the proposed method is efficient with the two-step technique.
基金Supported by the Scientific Research Project of Education Department of Hunan Province(Grant No.21C0837).
文摘A partition reduction method is used to obtain new upper bounds for the inverses of H-matrices and S-strictly diagonally dominant(S-SDD)matrices.The estimates are expressed via the determinants of third order matrices.Numerical experiments with various random matrices show that they are stable and better than the estimates presented in literatures.We use these upper bounds to improve known error estimates for linear complementarity problems with H-matrices and S-SDD matrices.
基金supported by National Natural Science Foundation of China (No. 10771120)
文摘This paper addresses the generalized linear complementarity problem （GLCP） over a polyhedral cone. To solve the problem, we first equivalently convert the problem into an affine variational inequalities problem over a closed polyhedral cone, and then propose a new type of method to solve the GLCP based on the error bound estimation. The global and R-linear convergence rate is established. The numerical experiments show the efficiency of the method.
文摘In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient projection method is given for solving the stochastic generalized linear complementarity problems. The global convergence of the conjugate gradient projection method is proved and the related numerical results are also reported.
基金Supported by the National Natural Science Foun dation of China(70371032)the Doctoral Educational Foundation 0f China of the Ministry of Education(20020486035)
文摘Feasible-interior-point algorithms start from a strictly feasible interior point,but infeassible-interior-point algorithms just need to start from an arbitrary positive point,we give a potential reduction algorithm from an infeasible-starting-point for a class of non-monotone linear complementarity problem.Its polynomial complexity is analyzed.After finite iterations the algorithm produces an approximate solution of the problem or shows that there is no feasible optimal solution in a large region.
文摘A one_step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so_called aggregation function. The proposed algorithm has the following good features: (ⅰ) It solves only one linear system of equations and does only one line search at each iteration; (ⅱ) It is well_defined for the vertical linear complementarity problem with vertical block P 0 matrix and any accumulation point of iteration sequence is its solution.Moreover, the iteration sequence is bounded for the vertical linear complementarity problem with vertical block P 0+R 0 matrix; (ⅲ) It has both global linear and local quadratic convergence without strict complementarity. Many existing smoothing Newton methods do not have the property (ⅲ).
基金Supported by the National Natural Science Foundation of China(71071119)
文摘This paper proposes an infeasible interior-point algorithm for linear complementarity problem with full-Newton steps.The main iteration consists of a feasibility step and several centrality steps.No more than O（n log（n /ε））iterations are required for getting ε-solution of the problem at hand,which coincides with the best-known bound for infeasible interior-point algorithms.
基金supported by the Natural Science Foundation of Hubei Province of China(2008CDZ047)
文摘It has been shown in various papers that most interior-point algorithms for linear optimization and their analysis can be generalized to P_＊（κ） linear complementarity problems.This paper presents an extension of the recent variant of Mehrotra＇s second order algorithm for linear optimijation.It is shown that the iteration-complexity bound of the algorithm is O（4κ ＋ 3）√14κ ＋ 5 nlog（x0）Ts0/ε,which is similar to that of the corresponding algorithm for linear optimization.
基金supported by the National Natural Science Foundation of China (Grant No.10771133)the Shanghai Pujiang Program (Grant No.06PJ14039)
文摘A polynomial interior-point algorithm is presented for monotone linear complementarity problem （MLCP） based on a class of kernel functions with the general barrier term, which are called general kernel functions. Under the mild conditions for the barrier term, the complexity bound of algorithm in terms of such kernel function and its derivatives is obtained. The approach is actually an extension of the existing work which only used the specific kernel functions for the MLCP.
文摘A modified sequential linear programming algorithm is presented, whose subproblem is always solvable, for the extended linear complementarity problem (XLCP), the global convergence of the algorithm under assumption of X-row sufficiency or X-colunm monotonicity is proved. As a result, a sufficient condition for existence and boundedness of solution to the XLCP are obtained.
基金Supported by the Natural Science Foundation of Hubei Province(Grant No.2008CDZ047)
文摘Mehrotra-type predictor-corrector algorithm, as one of most efficient interior point methods, has become the backbones of most optimization packages. Salahi et al. proposed a cut strategy based algorithm for linear optimization that enjoyed polynomial complexity and maintained its efficiency in practice. We extend their algorithm to P. （~） linear complementar- ity problems. The way of choosing corrector direction for our algorithm is different from theirs： The new algorithm has been proved to have an O（（1 ＋ 4k）（17 ＋ 19k）√1＋2kn 3/2 log（x0）Ts0/ε） worst case iteration complexity bound. An numerical experiment verifies the feasibility of the new algorithm.
基金Supported by the NNSF of China(11071041, 11171257)
文摘The extended linear complementarity problem(denoted by ELCP) can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed CHKS smoothing function, the ELCP is approximated by a family of parameterized smooth equations. A one-step smoothing Newton method is designed for solving the ELCP. The proposed algorithm is proved to be globally convergent under suitable assumptions.
基金This work is supported by the National Natural Science Foundation of China with No.11461046the Natural Science Foundation of Jiangxi Province of China with Nos.20181ACB20001 and 20161ACB21005.
文摘In this paper,by means of constructing the linear complementarity problems into the corresponding absolute value equation,we raise an iteration method,called as the nonlinear lopsided HSS-like modulus-based matrix splitting iteration method,for solving the linear complementarity problems whose coefficient matrix in R^(n×n)is large sparse and positive definite.From the convergence analysis,it is appreciable to see that the proposed method will converge to its accurate solution under appropriate conditions.Numerical examples demonstrate that the presented method precede to other methods in practical implementation.
基金supported by the National Natural Science Foundation of China(Nos.12171105,11271206)the Fundamental Research Funds for the Central Universities(No.FRF-DF-19-004).
文摘This paper considers semidefinite relaxation for linear and nonlinear complementarity problems.For some particular copositive matrices and tensors,the existence of a solution for the corresponding complementarity problems is studied.Under a general assumption,we show that if the solution set of a complementarity problem is nonempty,then we can get a solution by the semidefinite relaxation method;while if it does not have a solution,we can obtain a certificate for the infeasibility.Some numerical examples are given.
基金supported by the National Natural Science Foundation of China(Nos.11961082,12071159 and U1811464).
文摘In this paper,we discuss the perturbation analysis of the extended vertical linear complementarity problem(EVLCP).Under the assumption of the row W-property,we derive several absolute and relative perturbation bounds of EVLCP,which extend some existing results.Several numerical examples are given to show the proposed bounds.
文摘Recently, we have proposed an iterative projection and contraction (PC) method for a class of linear complementarity problems (LCP)([4]). The method was showed to be globally convergent, but no statement could be made about the rate of convergence. In this paper, we develop a modified globally linearly convergent PC method for linear complementarity problems. Both the method and the convergence proofs are very simple. The method can also be used to solve some linear variational inequalities. Several computational experiments are presented to indicate that the method is surprising good for solving some known difficult problems.
基金Subsidized by The Special Funds For Major State Basic Research Projects G1999032803.
文摘Asynchronous parallel multisplitting relaxation methods for solving large sparse linear complementarity problems are presented, and their convergence is proved when the system matrices are H-matrices having positive diagonal elements. Moreover, block and multi-parameter variants of the new methods, together with their convergence properties, are investigated in detail. Numerical results show that these new methods can achieve high parallel efficiency for solving the large sparse linear complementarity problems on multiprocessor systems.
