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 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.

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 On the complex complementarity problem

1973 ◽  
Vol 9 (2) ◽  
pp. 249-257 ◽  
Author(s):  
Bertram Mond
Keyword(s):  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Convex Cone ◽  
Complex Space ◽  
Linear Complementarity ◽  
Polar Cone ◽  
Polyhedral Convex ◽  
Complex Linear ◽  
Image Position

The complex linear complementarity problem considered here is the following: Find z such thatwhere S is a polyhedral convex cone in Cp, S* the polar cone, M ∈ Cp×p and q ∈ Cp.Generalizing earlier results in real and complex space, it is shown that if M satisfies RezHMz ≥ 0 for all z ∈ Cp and if the set satisfying Mz + q ∈ S*, z ∈ S is not empty, then a solution to the complex linear complementarity problem exists. If RezHMz > 0 unless z = 0, then a solution to this problem always exists.

scholarly journals Long step homogeneous interior point algorithm for the p* nonlinear complementarity problems

2002 ◽  
Vol 12 (1) ◽  
pp. 17-48
Author(s):  
Goran Lesaja
Keyword(s):  
Global Convergence ◽  
Linear Complementarity Problem ◽  
Interior Point ◽  
Complementarity Problem ◽  
Complementarity Problems ◽  
Nonlinear Complementarity Problem ◽  
Linear Complementarity ◽  
Compact Set ◽  
Interior Point Algorithm ◽  
Nonlinear Complementarity

A P*-Nonlinear Complementarity Problem as a generalization of the P*-Linear Complementarity Problem is considered. We show that the long-step version of the homogeneous self-dual interior-point algorithm could be used to solve such a problem. The algorithm achieves linear global convergence and quadratic local convergence under the following assumptions: the function satisfies a modified scaled Lipschitz condition, the problem has a strictly complementary solution, and certain submatrix of the Jacobian is nonsingular on some compact set.

scholarly journals Existence theory for the complex nonlinear complementarity problem

1976 ◽  
Vol 14 (3) ◽  
pp. 417-423 ◽  
Author(s):  
J. Parida ◽  
B. Sahoo
Keyword(s):  
Existence Theorem ◽  
Unique Solution ◽  
Complementarity Problem ◽  
Nonlinear Complementarity Problem ◽  
Polyhedral Cone ◽  
Existence Theory ◽  
Polar Cone ◽  
Nonlinear Complementarity ◽  
Strongly Monotone ◽  
Image Position

The main result in this paper is an existence theorem for the following complex nonlinear complementarity problem: find z such thatwhere S is a polyhedral cone in Cn, S* the polar cone, and g is a mapping from Cn into itself. It is shown that the above problem has a unique solution if the mapping g is continuous and strongly monotone on the polyhedral cone S.

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 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 A nonlinear complementarity problem in mathematical programming in Hilbert space

1979 ◽  
Vol 20 (2) ◽  
pp. 233-236 ◽  
Author(s):  
Sribatsa Nanda ◽  
Sudarsan Nanda
Keyword(s):  
Hilbert Space ◽  
Complementarity Problem ◽  
Existence And Uniqueness ◽  
Nonlinear Complementarity Problem ◽  
Closed Convex Cone ◽  
Existence And Uniqueness Theorem ◽  
Polar Cone ◽  
Nonlinear Complementarity ◽  
Banach Contraction ◽  
The Banach Contraction Principle

In this paper we prove the following existence and uniqueness theorem for the nonlinear complementarity problem by using the Banach contraction principle. If T: K → H is strongly monotone and lipschitzian with k2 < 2c < k2+1, then there is a unique y ∈ K, such that Ty ∈ K* and (Ty, y) = 0 where H is a Hilbert space, K is a closed convex cone in H, and K* the polar cone.

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