The paper presents concrete realizations of quasi-Newton methods for solving several standard problems including complementarity problems, special variational inequality problems, and the Karush–Kuhn–Tucker (KKT) system of nonlinear programming. A new approximation idea is introduced in this paper. The Q-superlinear convergence of the Newton method and the quasiNewton method are established und...

We discuss a nonlinear multigrid method for a linear complementarity problem. The convergence is improved by a recombination of iterants. The problem under consideration deals with option pricing from mathematical finance. Linear complementarity problems arise from so-called American-style options. A 2D convectiondiffusion type operator is discretized with the help of second order upwind discre...

In this paper, an algorithm for solving a Mathematical Programming Problem with Complementarity (or Equilibrium) Constraints (MPEC) is introduced, which uses the active-set methodology while maintaining the complementarity restrictions throughout the procedure. Finite convergence of the algorithm to a stationary point of the MPEC is established under reasonable hypotheses. The algorithm can be ...

We study several properties of a homotopy solution path associated with continuous nonlinear quasi-monotone complementarity problems. We use Poincare-Bohn’s homotopy invariance theorem of degree to derive an alternative theorem. Based on this result, a sufficient condition is established to assure the existence and boundedness of this homotopy solution path. Our results provide a theoretical ba...

The strict feasibility plays an important role in the development of theory and algorithms of complementarity problems. In this paper, we establish suucient conditions to ensure the strict feasibility of a nonlinear complemen-tarity problem. Our analytical method, based on a newly introduced concept of-antifeasible sequence, can be viewed as a uniied approach in proving the existence of a stric...

1 The authors would like to thank two anonymous referees for their helpful suggestions leading to a improved presentation of the paper. Abstract. Two interior-point algorithms using a wide neighborhood of the central path are proposed to solve nonlinear P-complementarity problems. The proof of the polynomial complexity of the rst method requires that the problem satisses a scaled Lipschitz cond...

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...

Recently, J.-S. Chen and P. Tseng extended two merit functions for the nonlinear complementarity problem (NCP) and the semidefinite complementarity problem (SDCP) to the second-order cone commplementarity problem (SOCCP) and showed several favorable properties. In this paper, we extend a merit function for the NCP studied by Yamada, Yamashita, and Fukushima to the SOCCP and show that the SOCCP ...

We consider a regularization method for nonlinear complementarity problems with F being a P0-function which replaces the original problem with a sequence of the regularized complementarity problems. In this paper, this sequence of regularized complementarity problems are solved approximately by applying the generalized Newton method for an equivalent augmented system of equations, constructed b...

The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, in...