A QP-free constrained Newton-type method for variational inequality problems

Received January 5, 1997 / Revised version received November 19, 1997 Published online November 24, 1998     

A penalized Fischer-Burmeister NCP-function

We introduce a new NCP-function in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations. This new NCP-function turns out to have stronger theoretical properties than the widely used Fischer-Burmeister function and other NCP-functions suggested previously. Moreover, numerical experience indicates that a semismooth Newton method based on this new NCP-function performs considerably better than the corresponding method based on the Fischer-Burmeister function. Received: March 10, 1997 / Accepted: February 15, 2000?Published online May 12, 2000     

用半光滑牛顿法求解一般的凸光顺问题

本文讨论一般的凸光顺问题minF（y）：=∫a^b（｜D^k y｜）^2dt＋∑（i=1）^N ωi｜y（ti）-zi｜^2．其中，忌芝3而且可在闭凸集凡K（∪→）L2^k[a,b]．我们把该问题转化为半光滑方程组并给出一个求解该方程组的半光滑牛顿算法．最后证明算法的超线性收敛性并给出数值算例．     

Approximating Clarke’s subgradients of semismooth functions by divided differences

We show that the algorithm presented in an earlier paper by Studniarski (Numer. Math., 55:685–693, 1989) can be applied, after only a small modification, to approximate numerically Clarke’s subgradients of semismooth functions of two variables. Results of computational testing of this modified algorithm are also reported.      

Strong Semismoothness of the Fischer-Burmeister SDC and SOC Complementarity Functions

We show that the Fischer-Burmeister complementarity functions, associated to the semidefinite cone (SDC) and the second order cone (SOC), respectively, are strongly semismooth everywhere. Interestingly enough, the proof relys on a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix.The author’s research was partially supported by Grant R146-000-035-101 of National University of Singapore.The author’s research was partially supported by Grant R314-000-042/057-112 of National University of Singapore and a grant from the Singapore-MIT Alliance.Mathematics Subject Classification (2000): 90C33, 90C22, 65F15, 65F18     

New smooth C-functions for symmetric cone complementarity problems

In this paper, with the help of the Jordan-algebraic technique we introduce two new complementarity functions (C-functions) for symmetric cone complementary problems, and show that they are continuously differentiable and strongly semismooth everywhere. The work was partly supported by the National Natural Science Foundation of China (10671010, 70471002).     

Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones

Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x∈H, a spectral decomposition is introduced, and for any function f:R→R, we define a corresponding vector-valued function f^H(x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x∈H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.     

二次锥规划的光滑牛顿法

在光滑Fischer-Burmeister函数的基础上,本文给出了二次锥规划的一种新的光滑牛顿法.该方法所采用的系统不是等价于中心路径条件,而是等价于最优性条件本身.算法对初始点没有任何限制,且具有Q-二阶收敛速度.     

A New Class of Semismooth Newton-Type Methods for Nonlinear Complementarity Problems

We introduce a new, one-parametric class of NCP-functions. This class subsumes the Fischer function and reduces to the minimum function in a limiting case of the parameter. This new class of NCP-functions is used in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations. We present a detailed investigation of the properties of the equation operator, of the corresponding merit function as well as of a suitable semismooth Newton-type method. Finally, numerical results are presented for this method being applied to a number of test problems.     

On a new exponential iterative method for solving nonsmooth equations

In this paper, we propose a new distinctive version of a generalized Newton method for solving nonsmooth equations. The iterative formula is not the classic Newton type, but an exponential one. Moreover, it uses matrices from B‐differential instead of generalized Jacobian. We prove local convergence of the method and we present some numerical examples.