On the nonlinear implicit complementarity problem (original) (raw)
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On the nonlinear complementarity problem
Journal of Mathematical Analysis and Applications, 1987
We consider and study an algorithm for a new class of complementarity problems of finding u such that u 2 0. Tu+A(u)>O, (u, Tu+A(u))=O, where T is a continuous mapping and A is a diagonally nonlinear mapping from [w" into itself. Furthermore, it is proved that the approximate solution obtained by the iterative scheme converges to the exact solution. Special cases are also discussed.
Solvability of implicit complementarity problems
Mathematical and Computer Modelling, 2007
In this paper, we introduce some new notions of (α, γ)-exceptional family of elements (in short, (α, γ)-(EFE)) and (α, β, γ)-exceptional family of elements (in short, (α, β, γ)-(EFE)) for a pair of continuous functions involved in the implicit complementarity problem (in short, ICP). Based upon these notions and the topological degree theory, we studied the feasibility and strictly feasibility of (ICP) in R n and an infinite-dimensional Hilbert space H , respectively. As special cases, we obtain the feasibility and strictly feasibility of complementarity problems and partly answered the second open problem (P2) proposed by Isac [G. Isac, Exceptional families of elements, feasibility and complementarity,
Solving complementarity problems by means of anew smooth constrained nonlinear solverRoberto
1999
Given F : IR n ! IR m and a closed and convex set, the problem of nding x 2 IR n such that x 2 and F (x) = 0 is considered. For solving this problem an algorithm of Inexact-Newton type is de-ned. Global and local convergence proofs are presented. As a practical application, the Horizontal Nonlinear Complementarity Problem is introduced. It is shown that the Inexact-Newton algorithm can be applied to this problem. Numerical experiments are performed and commented.
Iterative methods for a class of complementarity problems
Journal of Mathematical Analysis and Applications, 1988
In this paper, we propose and study an algorithm for a new class of complementarity problems of finding u E R" such that u > 0, Tu + A(u) > 0; (u, Tu + A(u)) = 0, where T is a continuous mapping and A is a nonlinear transformation from R" into itself. It is proved that the approximate solution obtained from the iterative scheme converges to the exact solution. Several special cases are also discussed.
An iterative technique for generalized strongly nonlinear complementarity problems
Applied Mathematics Letters, 1999
In this paper, using the change of variables technique, we establish the equivalence between the generalized strongly nonlinear complementarity problems and the Wiener-Hopf equations, This equivalence is used to suggest a new iterative method for the complementarity problems. We also study the convergence analysis of the iterative method and discuss some special cases.
On the Resolution of the Generalized Nonlinear Complementarity Problem
SIAM Journal on Optimization, 2002
Minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem (GNCP) defined on a polyhedral cone. It is not necessary to calculate projections that complicate and sometimes even disable the implementation of algorithms for solving these kinds of problems. Theoretical results that relate stationary points of the function that is minimized to the solutions of the GNCP are presented. Perturbations of the GNCP are also considered, and results are obtained related to the resolution of GNCPs with very general assumptions on the data. These theoretical results show that local methods for box-constrained optimization applied to the associated problem are efficient tools for solving the GNCP. Numerical experiments are presented that encourage the use of this approach.
International Journal of Computer Mathematics, 2006
Nonlinear complementarity problems (NCPs) are often solved by iterative methods based on a generalization of the classical Newton method and its modifications for smooth equations. We consider a method with modification of the right-hand side vector for a reformulation of the semi-smooth equation arising from NCPs and prove local convergence of the proposed method. Some numerical results are presented.
An iterative scheme for generalized mildly nonlinear complementarity problems
Applied Mathematics Letters, 1999
We use the change of variables technique to establish the equivalence between the generalized mildly nonlinear complementarity problems and the Wiener-Hopf equations. This equivalence is used to suggest a new iteratiX, e method for the complementarity problems. We also study the convergence analysis of the iterative method and discuss some special cases.
On the existence and uniqueness of solutions in nonlinear complementarity theory
Mathematical Programming, 1977
A complementarity problem is said to be globally uniquely solvable (GUS) if it has a unique solution, and this property will not change, even if any constant term is added to the mapping generating the problem. A characterization of the GUS property which generalizes a basic theorem in linear complementarity theory is given. Known sufficient conditions given by Cottle, Karamardian, and MorC for the nonlinear case are also shown to be generalized. In particular, several open questions concerning Cottle's condition are settled and a new proof is given for the sufficiency of this condition. A simple characterization for the two-dimensional case and a necessary condition for the n-dimensional case are also given. * T h e research described in this paper was carried out while N. Megiddo was visiting Tokyo lnstitute of Technology under a Fellowship of the Japan Society for the Promotion of Science.
Numerical validation of solutions of complementarity The nonlinear case
Numerische Mathematik, 2002
This paper proposes a validation method for solutions of nonlinear complementarity problems. The validation procedure performs a computational test. If the result of the test is positive, then it is guaranteed that a given multi-dimensional interval either includes a solution or excludes all solutions of the nonlinear complementarity problem.