A smoothing Newton method for the second-order cone complementarity problem

In this paper we introduce a new smoothing function and show that it is coercive under suitable assumptions. Based on this new function, we propose a smoothing Newton method for solving the second-order cone complementarity problem (SOCCP). The proposed algorithm solves only one linear system of equations and performs only one line search at each iteration. It is shown that any accumulation point of the iteration sequence generated by the proposed algorithm is a solution to the SOCCP. Furthermore, we prove that the generated sequence is bounded if the solution set of the SOCCP is nonempty and bounded. Under the assumption of nonsingularity, we establish the local quadratic convergence of the algorithm without the strict complementarity condition. Numerical results indicate that the proposed algorithm is promising.     

A smoothing Levenberg-Marquardt algorithm for solving a class of stochastic linear complementarity problem

In this paper, it is considered for a class of stochastic linear complementarity problems (SLCPs) with finitely many elements. A smoothing Levenberg-Marquardt algorithm is proposed for solving the SLCP. Under suitable conditions, the global convergence and local quadratic convergence of the proposed algorithm is given. Some numerical results are reported in this paper, which confirms the good theoretical properties of the proposed algorithm.     

A smoothing inexact Newton method for P 0 nonlinear complementarity problem

We first propose a new class of smoothing functions for the nonlinear 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 P ₀ nonlinear complementarity problem. The global convergence and local superlinear convergence are established. Preliminary numerical results indicate the feasibility and efficiency of the algorithm.     

A smoothing Newton algorithm for solving the monotone second-order cone complementarity problems

In this paper, a new smoothing function is given by smoothing the symmetric perturbed Fischer-Burmeister function. Based on this function, a smoothing Newton algorithm is proposed for solving the monotone second-order cone complementarity problems. The global and local quadratic convergence results of the algorithm are established under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool which is extensively used in our analysis. Numerical results indicate that the proposed algorithm is effective.     

A smoothing Newton method for solving a class of stochastic linear complementarity problems

In this paper, we consider a class of stochastic linear complementarity problems (SLCPs) with finitely many elements. We present a smoothing Newton algorithm for solving the SLCP. Under suitable conditions, we obtain the global convergence and locally quadratic convergence of the proposed algorithm. Some numerical results are reported in this paper, which confirm the good theoretical properties of the proposed algorithm.     

A new smoothing and regularization Newton method for the symmetric cone complementarity problem

Based on a regularized Chen–Harker–Kanzow–Smale (CHKS) smoothing function, we propose a new smoothing and regularization Newton method for solving the symmetric cone complementarity problem. By using the theory of Euclidean Jordan algebras, we establish the global and local quadratic convergence of the method on certain assumptions. The proposed method uses a nonmonotone line search technique which includes the usual monotone line search as a special case. In addition, our method treats both the smoothing parameter \(\mu \) and the regularization parameter \(\varepsilon \) as independent variables. Preliminary numerical results are reported which indicate that the proposed method is effective.     

A smoothing inexact Newton method for nonlinear complementarity problems

In this article, we propose a new smoothing inexact Newton algorithm for solving nonlinear complementarity problems (NCP) base on the smoothed Fischer-Burmeister function. In each iteration, the corresponding linear system is solved only approximately. The global convergence and local superlinear convergence are established without strict complementarity assumption at the NCP solution. Preliminary numerical results indicate that the method is effective for large-scale NCP.     

A new modified one-step smoothing Newton method for solving the general mixed complementarity problem

In last decades, there has been much effort on the solution and the analysis of the mixed complementarity problem (MCP) by reformulating MCP as an unconstrained minimization involving an MCP function. In this paper, we propose a new modified one-step smoothing Newton method for solving general (not necessarily P₀) mixed complementarity problems based on well-known Chen-Harker-Kanzow-Smale smooth function. Under suitable assumptions, global convergence and locally superlinear convergence of the algorithm are established.     

A variable dimension algorithm for the linear complementarity problem

A variable dimension algorithm is presented for the linear complementarity problems – Mz = q; s,z 0; s _i z _i = 0 fori = 1,2, ,n. The algorithm solves a sequence of subproblems of different dimensions, the sequence being possibly nonmonotonic in the dimension of the subproblem solved. Every subproblem is the linear complementarity problem defined by a leading principal minor of the matrixM. Index-theoretic arguments characterize the points at which nonmonotonic behavior occurs.     

A parallel algorithm for solving the linear complementarity problem

The concept of multitasking mathematical programs is discussed, and an application of multitasking to the multiple-cost-row linear programming problem is considered. Based on this, an algorithm for solving the Linear Complementarity Problem (LCP) in parallel is presented. A variety of computational results are presented using this multitasking approach on the CRAY X-MP/48. These results were obtained for randomly generated LCP's where thenxn dense matrixM has no special properties (hence, the problem is NP-hard). based on these results, an average time performance ofO(n ⁴) is observed.     

A predictor-corrector smoothing Newton method for symmetric cone complementarity problems

In this paper, we present a predictor-corrector smoothing Newton method for solving nonlinear symmetric cone complementarity problems (SCCP) based on the symmetrically perturbed smoothing function. Under a mild assumption, the solution set of the problem concerned is just nonempty, we show that the proposed algorithm is globally and locally quadratic convergent. Also, the algorithm finds a maximally complementary solution to the SCCP. Numerical results for second order cone complementarity problems (SOCCP), a special case of SCCP, show that the proposed algorithm is effective.     

One-step smoothing Newton method for solving the mixed complementarity problem with a function

The mixed complementarity problem (denote by MCP(F)) can be reformulated as the solution of a smooth system of equations. In the paper, based on a perturbed mid function, we propose a new smoothing function, which has an important property, not satisfied by many other smoothing function. The existence and continuity of a smooth path for solving the mixed complementarity problem with a P₀ function are discussed. Then we presented a one-step smoothing Newton algorithm to solve the MCP with a P₀ function. The global convergence of the proposed algorithm is verified under mild conditions. And by using the smooth and semismooth technique, the rate of convergence of the method is proved under some suitable assumptions.     

A new class of smoothing functions and a smoothing Newton method for complementarity problems

In this paper, we introduce a new class of smoothing functions, which include some popular smoothing complementarity functions. We show that the new smoothing functions possess a system of favorite properties. The existence and continuity of a smooth path for solving the nonlinear complementarity problem (NCP) with a P ₀ function are discussed. The Jacobian consistency of this class of smoothing functions is analyzed. Based on the new smoothing functions, we investigate a smoothing Newton algorithm for the NCP and discuss its global and local superlinear convergence. Some preliminary numerical results are reported.     

Basic-set algorithm for a generalized linear complementarity problem

In this paper, the authors develop a new direct method for the solution of a BLCP, that is, a linear complementarity problem (LCP) with upper bounds, when its matrix is a symmetric or an unsymmetricP-matrix. The convergence of the algorithm is established by extending Murty's principal pivoting method to an LCP which is equivalent to the BLCP. Computational experience with large-scale BLCPs shows that the basic-set method can solve efficiently large-scale BLCPs with a symmetric or an unsymmetricP-matrix.     

Improved smoothing Newton methods for symmetric cone complementarity problems

There recently has been much interest in smoothing Newton method for solving nonlinear complementarity problems. We extend such method to symmetric cone complementarity problems (SCCP). In this paper, we first investigate a one-parametric class of smoothing functions in the context of symmetric cones, which contains the Fischer–Burmeister smoothing function and the CHKS smoothing function as special cases. Then we propose a smoothing Newton method for the SCCP based on the one-parametric class of smoothing functions. For the proposed method, besides the classical step length, we provide a new step length and the global convergence is obtained. Finally, preliminary numerical results are reported, which show the effectiveness of the two step lengthes in the algorithm and provide efficient domains of the parameter for the complementarity problems.     

圆锥规划问题的光滑牛顿方法

给出求解圆锥规划问题的一种新光滑牛顿方法.基于圆锥互补函数的一个新光滑函数,将圆锥规划问题转化成一个非线性方程组,然后用光滑牛顿方法求解该方程组.该算法可从任意初始点开始,且不要求中间迭代点是内点.运用欧几里得代数理论,证明算法具有全局收敛性和局部超线性收敛速度.数值算例表明算法的有效性.     

A modified damped Newton method for linear complementarity problems

We present a modified damped Newton method for solving large sparse linear complementarity problems, which adopts a new strategy for determining the stepsize at each Newton iteration. The global convergence of the new method is proved when the system matrix is a nondegenerate matrix. We then apply the matrix splitting technique to this new method, deriving an inexact splitting method for the linear complementarity problems. The global convergence of the resulting inexact splitting method is proved, too. Numerical results show that the new methods are feasible and effective for solving the large sparse linear complementarity problems.