scholarly journals An existence theorem for the generalized complementarity problem

1978 ◽  
Vol 19 (1) ◽  
pp. 51-58
Author(s):  
J. Parida ◽  
B. Sahoo
Keyword(s):  
Complementarity Problem ◽  
Convex Cone ◽  
Existence Theorems ◽  
Nonlinear Complementarity Problem ◽  
Closed Convex Cone ◽  
Coercivity Conditions ◽  
Existence Of A Solution ◽  
New Class ◽  
Image Position ◽  
Class Of Functions

Given a closed, convex cone S, in Rn, its polar S* and a mapping g from Rn into itself, the generalized nonlinear complementarity problem is to find a z ∈ Rn such thatMany existence theorems for the problem have been established under varying conditions on g. We introduce new mappings, denoted by J(S)-functions, each of which is used to guarantee the existence of a solution to the generalized problem under certain coercivity conditions on itself. A mapping g:S → Rn is a J(S)-function ifimply that z = 0. It is observed that the new class of functions is a broader class than the previously studied ones.

scholarly journals A complex nonlinear complementarity problem

1978 ◽  
Vol 19 (3) ◽  
pp. 437-444 ◽  
Author(s):  
Sribatsa Nanda ◽  
Sudarsan Nanda
Keyword(s):  
Continuous Function ◽  
Complementarity Problem ◽  
Convex Cone ◽  
Existence And Uniqueness ◽  
Nonlinear Complementarity Problem ◽  
Closed Convex Cone ◽  
Uniqueness Of Solutions ◽  
Existence Of A Solution ◽  
Polar Cone ◽  
Nonlinear Complementarity

In this paper we study the existence and uniqueness of solutions for the following complex nonlinear complementarity problem: find z ∈ S such that g(z) ∈ S* and re(g(z), z) = 0, where S is a closed convex cone in Cn, S* the polar cone, and g is a continuous function from Cn into itself. We show that the existence of a z ∈ S with g(z) ∈ int S* implies the existence of a solution to the nonlinear complementarity problem if g is monotone on S and the solution is unique if g is strictly monotone. We also show that the above problem has a unique solution if the mapping g is strongly monotone on S.

scholarly journals A note on generalised linear complementarity problems

1978 ◽  
Vol 18 (2) ◽  
pp. 161-168 ◽  
Author(s):  
J. Parida ◽  
B. Sahoo
Keyword(s):  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Closed Convex Cone ◽  
Arbitrary Vector ◽  
Existence Of A Solution ◽  
Polar Cone ◽  
New Class ◽  
Class A ◽  
Class Of Matrices ◽  
Image Position

Given an n × n matrix A, an n-dimensional vector q, and a closed, convex cone S of Rn, the generalized linear complementarity problem considered here is the following: find a z ∈ Rn such thatwhere s* is the polar cone of S. The existence of a solution to this problem for arbitrary vector q has been established both analytically and constructively for several classes of matrices A. In this note, a new class of matrices, denoted by J, is introduced. A is a J-matrix ifThe new class can be seen to be broader than previously studied classes. We analytically show that for any A in this class, a solution to the above problem exists for arbitrary vector q. This is achieved by using a result on variational inequalities.

scholarly journals On the complex nonlinear complementary problem

1976 ◽  
Vol 14 (1) ◽  
pp. 129-136 ◽  
Author(s):  
J. Parida ◽  
B. Sahoo
Keyword(s):  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Nonlinear Complementarity Problem ◽  
Linear Complementarity ◽  
Feasible Point ◽  
Existence Of A Solution ◽  
Polar Cone ◽  
Nonlinear Complementarity ◽  
Image Position ◽  
Nonlinear Mappings

The complex nonlinear complementarity problem considered here is the following: find z such thatwhere S is a polyhedral cone in Cn, S* the polar cone, and g is a mapping from Cn into itself. We study the extent to which the existence of a z ∈ S with g(z) ∈ S* (feasible point) implies the existence of a solution to the nonlinear complementarity problem, and extend, to nonlinear mappings, known results in the linear complementarity problem on positive semi-definite matrices.

scholarly journals Global Error Bound Estimation for the Generalized Nonlinear Complementarity Problem over a Closed Convex Cone

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Hongchun Sun ◽  
Yiju Wang
Keyword(s):  
Error Bound ◽  
Complementarity Problem ◽  
Convex Cone ◽  
Nonlinear Complementarity Problem ◽  
Global Error ◽  
Closed Convex Cone ◽  
Global Error Bound ◽  
Nonlinear Complementarity ◽  
Bound Estimation

The global error bound estimation for the generalized nonlinear complementarity problem over a closed convex cone (GNCP) is considered. To obtain a global error bound for the GNCP, we first develop an equivalent reformulation of the problem. Based on this, a global error bound for the GNCP is established. The results obtained in this paper can be taken as an extension of previously known results.

On an application of the complex nonlinear complementarity problem

1976 ◽  
Vol 15 (1) ◽  
pp. 141-148 ◽  
Author(s):  
J. Parida ◽  
B. Sahoo
Keyword(s):  
Minimization Problem ◽  
Complementarity Problem ◽  
Complex Space ◽  
Nonlinear Complementarity Problem ◽  
Polyhedral Cone ◽  
Convex Minimization ◽  
Convex Minimization Problem ◽  
Polyhedral Cones ◽  
Existence Of A Solution ◽  
Nonlinear Complementarity

A theorem on the existence of a solution under feasibility assumptions to a convex minimization problem over polyhedral cones in complex space is given by using the fact that the problem of solving a convex minimization program naturally leads to the consideration of the following nonlinear complementarity problem: given g: Cn → Cn, find z such that g(z) ∈ S*, z ∈ S, and Re〈g(z), z〉 = 0, where S is a polyhedral cone and S* its polar.

scholarly journals Generalized AOR Method for Solving Absolute Complementarity Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Javed Iqbal ◽  
Khalida Inayat Noor ◽  
E. Al-Said
Keyword(s):  
Complementarity Problem ◽  
Complementarity Problems ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Equivalent Formulation ◽  
Existence Of A Solution ◽  
Absolute Value ◽  
New Class ◽  
Aor Method ◽  
The Absolute

We introduce and consider a new class of complementarity problems, which is called the absolute value complementarity problem. We establish the equivalence between the absolute complementarity problems and the fixed point problem using the projection operator. This alternative equivalent formulation is used to discuss the existence of a solution of the absolute value complementarity problem. A generalized AOR method is suggested and analyzed for solving the absolute the complementarity problems. We discuss the convergence of generalized AOR method for theL-matrix. Several examples are given to illustrate the implementation and efficiency of the method. Results are very encouraging and may stimulate further research in this direction.

scholarly journals A linear complementarity problem involving a subgradient

1988 ◽  
Vol 37 (3) ◽  
pp. 345-351 ◽  
Author(s):  
J. Parida ◽  
A. Sen ◽  
A. Kumar
Keyword(s):  
Convex Function ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Nonlinear Complementarity Problem ◽  
Linear Complementarity ◽  
Existence Of A Solution ◽  
Nonlinear Complementarity ◽  
The Matrix ◽  
Square Matrix

A linear complementarity problem, involving a given square matrix and vector, is generalised by including an element of the subdifferential of a convex function. The existence of a solution to this nonlinear complementarity problem is shown, under various conditions on the matrix. An application to convex nonlinear nondifferentiable programs is presented.

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