In this paper, we propose a regularized version of the generalized NCP-function proposed by Hu, Huang and Chen [J. Comput. Appl. Math., 230 (2009), pp. 69-82]. Based on this regularized function, we propose a semismooth Newton method for solving nonlinear complementarity problems, where a non-monotone line search scheme is used. In particular, we show that the proposed non-monotone method is globally and locally superlinearly convergent under suitable assumptions. We test the proposed method by solving the test problems from MCPLIB. Numerical experiments indicate that this algorithm has better numerical performance in the case of p = 5 and Θ ∈ [0.25,075] than other cases.

A multiplicative Schwarz iteration algorithm is presented for solving the finite-dimensional nonlinear complementarity problem with an M-function. The monotone convergence of the iteration algorithm is obtained with special choices of initial values. Moreover, by applying the concept of weak regular splitting, the weighted max-norm bound is derived for the iteration errors.

In this work we study the nonlinear complementarity problem on thenonnegative orthant. This is done by approximating its equivalentvariational-inequality-formulation by a sequence of variationalinequalities with nested compact domains. This approach yieldssimultaneously existence, sensitivity, and stability results. Byintroducing new classes of functions and a suitable metric forperforming the approximation, we provide bounds for the asymptoticset of the solution set and coercive existence results, which extendand generalize most of the existing ones from the literature. Suchresults are given in terms of some sets called coercive existencesets, which we also employ for obtaining new sensitivity andstability results. Topological properties of thesolution-set-mapping and bounds for it are also established.Finally, we deal with the piecewise affine case.

We describe the solution of a bound constrained convex quadratic problem with limited memory resources. The problem arises from physical simulations occurring within video games. The motivating problem is outlined, along with a simple interior point approach for its solution. Various linear algebra issues arising in the implementation are explored, including preconditioning, ordering and a number of ways of solving an equivalent augmented system. Alternative approaches are briefly surveyed, and some recommendations for solving these types of problems are given.

We consider a generalized proximal point method (GPPA) forsolving the nonlinear complementarity problem with monotone operators inRn . It differs from the classical proximal point method discussedby Rockafellar for the problem of finding zeroes of monotone operatorsin the use of generalized distances, called φ-divergences,instead of the Euclidean one. These distances play not only aregularization role but also a penalization one, forcing the sequencegenerated by the method to remain in the interior of the feasible set,so that the method behaves like an interior point one. Under appropriateassumptions on the φ-divergence and the monotone operator weprove that the sequence converges if and only if the problem hassolutions, in which case the limit is a solution. If the problem doesnot have solutions, then the sequence is unbounded. We extend previousresults for the proximal point method concerning convex optimizationproblems.