Smoothing Newton algorithm for symmetric cone complementarity problems based on a one-parametric class of smoothing functions

In this paper, we introduce a one-parametric class of smoothing functions in the context of symmetric cones which contains two symmetric perturbed smoothing functions as special cases, and show that it is coercive under suitable assumptions. Based on this class of smoothing functions, a smoothing Newton algorithm is extended to solve the complementarity problems over symmetric cones, and it is proved that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool which is extensively used in our analysis. Preliminary numerical results for randomly generated second-order cone programs and several practical second-order cone programs from the DIMACS library are reported.     

Convergence of a smoothing algorithm for symmetric cone complementarity problems with a nonmonotone line search

In this paper, we propose a smoothing algorithm for solving the monotone symmetric cone complementarity problems (SCCP for short) with a nonmonotone line search. We show that the nonmonotone algorithm is globally convergent under an assumption that the solution set of the problem concerned is nonempty. Such an assumption is weaker than those given in most existing algorithms for solving optimization problems over symmetric cones. We also prove that the solution obtained by the algorithm is a maximally complementary solution to the monotone SCCP under some assumptions. This work was supported by National Natural Science Foundation of China (Grant Nos. 10571134, 10671010) and Natural Science Foundation of Tianjin (Grant No. 07JCYBJC05200)     

The convergence of a modified smoothing-type algorithm for the symmetric cone complementarity problem

The symmetric cone complementarity problem (denoted by SCCP) is a broad class of optimization problems, which contains the semidefinite complementarity problem, the second-order cone complementarity problem, and the nonlinear complementarity problem. In this paper we first extend the smoothing function proposed by Huang et al. (Sci. China 44:1107–1114, 2001) for the nonlinear complementarity problem to the context of symmetric cones and show that it is coercive under suitable assumptions. Based on this smoothing function, a smoothing-type algorithm, which is a modified version of the Qi-Sun-Zhou method (Qi et al. in Math. Program. 87:1–35, 2000), is proposed for solving the SCCP. By using the theory of Euclidean Jordan algebras, we prove that the proposed algorithm is globally and locally quadratically convergent under suitable assumptions. Preliminary numerical results for some second-order cone complementarity problems are reported which indicate that the proposed algorithm is effective.     

基于尺度中心路径的求解SCLP的非单调光滑牛顿算法

基于CHKS光滑函数的修改性版本,该文提出了一个带有尺度中心路径的求解对称锥线性规划(SCLP)的非单调光滑牛顿算法.通过应用欧氏若当代数理论,在适当的假设下,证明了该算法是全局收敛和超线性收敛的.数值结果表明了算法的有效性.     

Convergence of a non-interior continuation algorithm for the monotone SCCP

It is well known that the symmetric cone complementarity problem（SCCP） is a broad class of optimization problems which contains many optimization problems as special cases.Based on a general smoothing function,we propose in this paper a non-interior continuation algorithm for solving the monotone SCCP.The proposed algorithm solves at most one system of linear equations at each iteration.By using the theory of Euclidean Jordan algebras,we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions.     

A Smoothing Newton Algorithm for a Class of Non-monotonic Symmetric Cone Linear Complementarity Problems

Recently, the study of symmetric cone complementarity problems has been a hot topic in the literature. Many numerical methods have been proposed for solving such a class of problems. Among them, the problems concerned are generally monotonic. In this paper, we consider symmetric cone linear complementarity problems with a class of non-monotonic transformations. A smoothing Newton algorithm is extended to solve this class of non-monotonic symmetric cone linear complementarity problems; and the algorithm is proved to be well-defined. In particular, we show that the algorithm is globally and locally quadratically convergent under mild assumptions. The preliminary numerical results are also reported.     

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.     

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.     

A family of new smoothing functions and?a?nonmonotone smoothing Newton method for?the?nonlinear complementarity problems

In this paper, based on a p-norm with p being any fixed real number in the interval (1,+??), we introduce a family of new smoothing functions, which include the smoothing symmetric perturbed Fischer function as a special case. We also show that the functions have several favorable properties. Based on the new smoothing functions, we propose a nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problems. The proposed algorithm only need to solve one linear system of equations. We show that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. Numerical experiments indicate that the method associated with a smaller p, for example p=1.1, usually has better numerical performance than the smoothing symmetric perturbed Fischer function, which exactly corresponds to p=2.     

Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets

This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.

     

A non-interior continuation algorithm for the CP based on a generalized smoothing function

Based on the generalized CP-function proposed by Hu et al. [S.L. Hu, Z.H. Huang, J.S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math. 230 (2009) 69-82], we introduce a smoothing function which is a generalization of several popular smoothing functions. By which we propose a non-interior continuation algorithm for solving the complementarity problem. The proposed algorithm only needs to solve at most one system of linear equations at each iteration. In particular, we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions. The preliminary numerical results demonstrate that the algorithm is effective.     

A new class of smoothing complementarity functions over symmetric cones

A popular approach to solving the complementarity problem is to reformulate it as an equivalent system of smooth equations via a smoothing complementarity function. In this paper, first we propose a new class of smoothing complementarity functions, which contains the natural residual smoothing function and the Fischer–Burmeister smoothing function for symmetric cone complementarity problems. Then we give some unified formulae of the Fréchet derivatives associated with Jordan product. Finally, the derivative of the new proposed class of smoothing complementarity functions is deduced over symmetric cones.     

Predictor-Corrector Smoothing Newton Method,Based on a New Smoothing Function,for Solving the Nonlinear Complementarity Problem with a P 0 Function

By smoothing a perturbed minimum function, we propose in this paper a new smoothing function. The existence and continuity of a smooth path for solving the nonlinear complementarity problem (NCP) with a P ₀ function are discussed. We investigate the boundedness of the iteration sequence generated by noninterior continuation/smoothing methods under the assumption that the solution set of the NCP is nonempty and bounded. Based on the new smoothing function, we present a predictor-corrector smoothing Newton algorithm for solving the NCP with a P ₀ function, which is shown to be globally linearly and locally superlinearly convergent under suitable assumptions. Some preliminary computational results are reported.     

A smoothing-type algorithm for solving nonlinear complementarity problems with a non-monotone line search

The smoothing-type algorithm has been successfully applied to solve various optimization problems. In general, the smoothing-type algorithm is designed based on some monotone line search. However, in order to achieve better numerical results, the non-monotone line search technique has been used in the numerical computations of some smoothing-type algorithms. In this paper, we propose a smoothing-type algorithm for solving the nonlinear complementarity problem with a non-monotone line search. We show that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are also reported.     

A smoothing-type algorithm for the second-order cone complementarity problem with a new nonmonotone line search

The second-order cone complementarity problem (denoted by SOCCP) can be effectively solved by smoothing-type algorithms, which in general are designed based on some monotone line search. In this paper, based on a new smoothing function of the Fischer–Burmeister function, we propose a smoothing-type algorithm for solving the SOCCP. The proposed algorithm uses a new nonmonotone line search scheme, which contains the usual monotone line search as a special case. Under suitable assumptions, we show that the proposed algorithm is globally and locally quadratically convergent. Some numerical results are reported which indicate the effectiveness of the proposed algorithm.     

A full-Newton step non-interior continuation algorithm for a class of complementarity problems

In this paper, we investigate a class of nonlinear complementarity problems arising from the discretization of the free boundary problem, which was recently studied by Sun and Zeng [Z. Sun, J. Zeng, A monotone semismooth Newton type method for a class of complementarity problems, J. Comput. Appl. Math. 235 (5) (2011) 1261–1274]. We propose a new non-interior continuation algorithm for solving this class of problems, where the full-Newton step is used in each iteration. We show that the algorithm is globally convergent, where the iteration sequence of the variable converges monotonically. We also prove that the algorithm is globally linearly and locally superlinearly convergent without any additional assumption, and locally quadratically convergent under suitable assumptions. The preliminary numerical results demonstrate the effectiveness of the proposed algorithm.     

A Superlinearly Convergent Implicit Smooth SQP Algorithm for Mathematical Programs with Nonlinear Complementarity Constraints

This paper discusses a special class of mathematical programs with nonlinear complementarity constraints, its goal is to present a globally and superlinearly convergent algorithm for the discussed problems. We first reformulate the complementarity constraints as a standard nonlinear equality and inequality constraints by making use of a class of generalized smoothing complementarity functions, then present a new SQP algorithm for the discussed problems. At each iteration, with the help of a pivoting operation, a master search direction is yielded by solving a quadratic program, and a correction search direction for avoiding the Maratos effect is generated by an explicit formula. Under suitable assumptions, without the strict complementarity on the upper-level inequality constraints, the proposed algorithm converges globally to a B-stationary point of the problems, and its convergence rate is superlinear.AMS Subject Classification: 90C, 49MThis work was supported by the National Natural Science Foundation (10261001) and the Guangxi Province Science Foundation (0236001, 0249003) of China.     

P_0函数非线性互补问题的一步非内点连续方法的收敛性

本文对于Ｐ０函数非线性互补问题提出了一个基于Ｋａｎｚｏｗ光滑函数的一步非内点连续方法,在适当的假设条件下,证明了方法的全局线性及局部二次收敛性．特别,在方法的全局线性收敛性的分析中,不需要假定非线性互补问题的函数的Ｊａｃｏｂｉ阵是Ｌｉｐｓｃｈｉｔｚ连续的．文献中为了得到非内点连续方法的全局线性收敛性,这一假定是被广泛使用的．本文提出的方法在每一次迭代只须解一个线性方程式组．     

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.     

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.