In this article, we consider an unconstrained minimization formulation of the nonlinear complementarity problem NCP(f) when the underlying functions are H-differentiable but not necessarily locally Lipschitzian or directionally differentiable. We show how, under appropriate regularity conditions on an H-differential of f , minimizing the merit function corresponding to f leads to a solution of ...

We introduce basic ideas of a nonsmooth Newton’s method and its application in solving semidefinite optimization (SDO) problems. In particular, the method can be used to solve both linear and nonlinear semidefinite complementarity problems. We also survey recent theoretical results in matrix functions and stability of SDO that are stemed from the research on the matrix form of the nonsmooth New...

Abstract. We study the uniform nonsingularity property recently proposed by the authors and present its applications to nonlinear complementarity problems over a symmetric cone. In particular, by addressing theoretical issues such as the existence of Newton directions, the boundedness of iterates and the nonsingularity of B-subdifferentials, we show that the non-interior continuation method pro...

We show that the gradient mapping of the squared norm of Fischer-Burmeister function is globally Lipschitz continuous and semismooth, which provides a theoretical basis for solving nonlinear second order cone complementarity problems via the conjugate gradient method and the semismooth Newton’s method.

The Kantorovich Theorem is a fundamental tool in nonlinear analysis which has been extensively used in classical numerical analysis. In this paper we show that it can also be used in analyzing interior point methods. We obtain optimal bounds for Newton’s method when relied upon in a path following algorithm for linear complementarity problems. Given a point z that approximates a point z(τ ) on ...

We present a local convergence analysis of generalized Newton methods for singular smooth and nonsmooth operator equations using adaptive constructs of outer inverses. We prove that for a solution x of F(x) = 0, there exists a ball S = S(x ; r), r > 0 such that for any starting point x 0 2 S the method converges to a solution x 2 S of ?F (x) = 0, where ? is a bounded linear operator that depend...

Combining nonmonotone trust region algorithms and PSO methods, we propose a mixed method for nonlinear complementarity problems. The iterative formula of k μ is very simple. When the determinant of k k B I λ + is very large or small, the PSO method will obtain a new point which gets us better astringency. Numerical test results show the effectiveness of this algorithm.

Recently Tseng [Merit function for semidefinite complementarity, Mathematical Programming, 83, pp. 159-185, 1998] extended a class of merit functions, proposed by Z. Luo and P. Tseng [A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems: State of the Art, pp. 204-225, 1997], for the nonlinear complementarity problem (NCP) to the s...

In this paper, we introduce a new class of generalized strongly set-valued nonlinear complementarity problems and construct new iterative algorithms. We show the existence of solutions for this kind of nonlinear complementarity problems and the convergence of iterative sequences generated by the algorithm. Our results extend some recent results in this field.

In this paper we develop a numerical approach to a fractional-order differential linear complementarity problem (LCP) arising in pricing European and American options under a geometric Lévy process. The (LCP) is first approximated by a penalized nonlinear fractional Black-Scholes (fBS) equation. To numerically solve this nonlinear (fBS), we use the horizontal method of lines to discretize the t...