One-step Smoothing Newton Method Research Articles

A new one-step smoothing Newton method for second-order cone programming

In this paper, we present a new one-step smoothing Newton method for solving the second-order cone programming (SOCP). Based on a new smoothing function of the well-known Fischer-Burmeister function, the SOCP is approximated by a family of parameterized smooth equations. Our algorithm solves only one system of linear equations and performs only one Armijo-type line search at each iteration. It can start from an arbitrary initial point and does not require the iterative points to be in the sets of strictly feasible solutions. Without requiring strict complementarity at the SOCP solution, the proposed algorithm is shown to be globally and locally quadratically convergent under suitable assumptions. Numerical experiments demonstrate the feasibility and efficiency of our algorithm.

A new one-step smoothing newton method for the second-order cone complementarity problem

In this paper, we present a new one-step smoothing Newton method for solving the second-order cone complementarity problem (SOCCP). Based on a new smoothing function, the SOCCP is approximated by a family of parameterized smooth equations. At each iteration, the proposed algorithm only need to solve one system of linear equations and perform only one Armijo-type line search. The algorithm is proved to be convergent globally and superlinearly without requiring strict complementarity at the SOCCP solution. Moreover, the algorithm has locally quadratic convergence under mild conditions. Numerical experiments demonstrate the feasibility and efficiency of the new algorithm. Copyright © 2010 John Wiley & Sons, Ltd.

A new one-step smoothing Newton method for nonlinear complementarity problem with -function

In this paper, nonlinear complementarity problem with P 0 -function is studied. Based on a new smoothing function, the problem is approximated by a family of parameterized smooth equations and we present a new one-step smoothing Newton method to solve it. At each iteration, the proposed method only need to solve one system of linear equations and perform one Armijo-type line search. The algorithm is proved to be convergent globally and superlinearly without requiring strict complementarity at the solution. Numerical experiments demonstrate the feasibility and efficiency of the new algorithm.

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.

One-step smoothing Newton method for solving the mixed complementarity problem with a [formula omitted] 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 P 0 function are discussed. Then we presented a one-step smoothing Newton algorithm to solve the MCP with a P 0 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.

Improved smoothing Newton methods for [formula omitted] nonlinear complementarity problems

In this paper, we consider a one-step smoothing Newton method for the solution of nonlinear complementarity problems with P 0 -functions. The proposed algorithm is based on the smoothing symmetric perturbed Fischer function, and it is proven to generate bounded iteration sequence and to possess strongly global and local convergence properties under weaker conditions. Preliminary numerical results indicate that the proposed algorithm is promising.

A variant smoothing Newton method for P0- NCP based on a new smoothing function

In this paper, we present a new one-step smoothing Newton method proposed for solving the non-linear complementarity problem with P 0 -function based on a new smoothing N C P -function. We adopt a variant merit function. Our algorithm needs only to solve one linear system of equations and perform one line search per iteration. It shows that any accumulation point of the iteration sequence generated by our algorithm is a solution of P 0 - N C P . Furthermore, under the assumption that the solution set is non-empty and bounded, we can guarantee at least one accumulation point of the generated sequence. Numerical experiments show the feasibility and efficiency of the algorithm.

A one-step smoothing Newton method for second-order cone programming

A new smoothing function for the second-order cone programming is given by smoothing the symmetric perturbed Fischer–Burmeister function. Based on this new function, a one-step smoothing Newton method is presented for solving the second-order cone programming. The proposed algorithm solves only one linear system of equations and performs only one line search at each iteration. This algorithm does not have restrictions regarding its starting point and is Q-quadratically convergent. Numerical results suggest the effectiveness of our algorithm.

The convergence of a one-step smoothing Newton method for [formula omitted]-NCP based on a new smoothing NCP-function

The nonlinear complementarity problem (denoted by NCP( F)) can be reformulated as the solution of a nonsmooth system of equations. By introducing a new smoothing NCP-function, the problem is approximated by a family of parameterized smooth equations. A one-step smoothing Newton method is proposed for solving the nonlinear complementarity problem with P 0 -function ( P 0 -NCP) based on the new smoothing NCP-function. The proposed algorithm solves only one linear system of equations and performs only one line search per iteration. Without requiring strict complementarity assumption at the P 0 -NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the algorithm has local quadratic convergence under mild conditions.

Global linear and quadratic one-step smoothing newton method for vertical linear complementarity problems

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 P0 matrix and any accumulation point of iteration sequence is its solution. Moreover, the iteration sequence is bounded for the vertical linear complementarily problem with vertical block P0+R0 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).

Global Linear and Quadratic One-step Smoothing Newton Method for P 0 -LCP

We propose a new smoothing Newton method for solving the P0-matrix linear complementarity problem (P0-LCP) based on CHKS smoothing function. Our algorithm solves only one linear system of equations and performs only one line search per iteration. It is shown to converge to a P0-LCP solution globally linearly and locally quadratically without the strict complementarity assumption at the solution. To the best of author's knowledge, this is the first one-step smoothing Newton method to possess both global linear and local quadratic convergence. Preliminary numerical results indicate that the proposed algorithm is promising.

Quadratic one-step smoothing Newton method for P0-LCP without strict complementarity

In this paper we propose a modified one-step smoothing Newton method for solving the P 0 linear complementarity problem ( P 0-LCP) based on Kanzow’s smoothing function. Our smoothing Newton method solves only one linear system of equations and performs only one line search at each iteration. It is proved that our proposed algorithm has global convergence and local quadratic convergence in absence of strict complementarity assumption at the P 0-LCP solution. Under weaker conditions, our convergence results are much stronger than many previous literatures.

Superlinear/quadratic one-step smoothing Newton method for P 0 -NCP without strict complementarity

We propose a modified one-step smoothing Newton method for solving nonlinear complementarity problems with P 0-function (P 0-NCP) based on CHKS smoothing function. Our smoothing Newton method solves only one linear system of equations and performs only one line search at each iteration. It is proved that our proposed algorithm has superlinear convergence in absence of strict complementarity assumption at the P 0-NCP solution. Under suitable conditions, the modified algorithm has global linear and local quadratic convergence. Compared to previous literatures, our algorithm has stronger convergence results under weaker conditions.