A note on generalised linear complementarity problems

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.

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On the complex nonlinear complementary problem

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700024898 ◽

1976 ◽

Vol 14 (1) ◽

pp. 129-136 ◽

Cited By ~ 8

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.

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An existence theorem for the generalized complementarity problem

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700008431 ◽

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.

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A complex nonlinear complementarity problem

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700008960 ◽

1978 ◽

Vol 19 (3) ◽

pp. 437-444 ◽

Cited By ~ 6

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.

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Minimising quadratic functionals over closed convex cones

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700027933 ◽

1989 ◽

Vol 39 (1) ◽

pp. 15-20 ◽

Cited By ~ 2

Author(s):

M. Seetharama Gowda

Keyword(s):

Complementarity Problems ◽

Real Hilbert Space ◽

Linear Complementarity Problems ◽

Linear Complementarity ◽

Convex Cones ◽

Closed Convex Cone ◽

Quadratic Functional ◽

Quadratic Functionals ◽

Finite Dimensional ◽

Bounded Below

In this article we show that, under suitable conditions a quadratic functional attains its minimum on a closed convex cone (in a finite dimensional real Hilbert space) whenever it is bounded below on the cone. As an application, we solve Generalised Linear Complementarity Problems over closed convex cones.

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Karamardian Matrices: An Analogue of Q-Matrices

Electronic Journal of Linear Algebra ◽

10.13001/ela.2021.5017 ◽

2021 ◽

Vol 37 (37) ◽

pp. 127-155

Author(s):

K.C. Sivakumar ◽

Sushmitha Parameswaran ◽

Megan Wendler

Keyword(s):

Null Space ◽

Zero Solution ◽

Linear Complementarity ◽

Range Space ◽

Systematic Treatment ◽

Solution Vector ◽

New Class ◽

Nonnegative Orthant ◽

Q Matrix ◽

Class Of Matrices

A real square matrix $A$ is called a $Q$-matrix if the linear complementarity problem LCP$(A,q)$ has a solution for all $q \in \mathbb{R}^n$. This means that for every vector $q$ there exists a vector $x$ such that $x \geq 0, y=Ax+q\geq 0$, and $x^Ty=0$. A well-known result of Karamardian states that if the problems LCP$(A,0)$ and LCP$(A,d)$ for some $d\in \mathbb{R}^n, d >0$ have only the zero solution, then $A$ is a $Q$-matrix. Upon relaxing the requirement on the vectors $d$ and $y$ so that the vector $y$ belongs to the translation of the nonnegative orthant by the null space of $A^T$, $d$ belongs to its interior, and imposing the additional condition on the solution vector $x$ to be in the intersection of the range space of $A$ with the nonnegative orthant, in the two problems as above, the authors introduce a new class of matrices called Karamardian matrices, wherein these two modified problems have only zero as a solution. In this article, a systematic treatment of these matrices is undertaken. Among other things, it is shown how Karamardian matrices have properties that are analogous to those of $Q$-matrices. A subclass of a recently introduced notion of $P_{\#}$-matrices is shown to possess the Karamardian property, and for this reason we undertake a thorough study of $P_{\#}$-matrices and make some fundamental contributions.

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On the complex complementarity problem

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700043148 ◽

1973 ◽

Vol 9 (2) ◽

pp. 249-257 ◽

Cited By ~ 5

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.

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A Fast Algorithm for Solving a Class of the Linear Complementarity Problem in a Finite Number of Steps

Abstract and Applied Analysis ◽

10.1155/2020/8881915 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8

Author(s):

Yamna Achik ◽

Asmaa Idmbarek ◽

Hajar Nafia ◽

Imane Agmour ◽

Youssef El foutayeni

Keyword(s):

Finite Number ◽

Linear Complementarity Problem ◽

Complementarity Problem ◽

Computation Time ◽

Linear Complementarity ◽

Major Drawback ◽

New Class ◽

Large Systems ◽

Class Of Matrices ◽

Number Of Iterations

The linear complementarity problem is receiving a lot of attention and has been studied extensively. Recently, El foutayeni et al. have contributed many works that aim to solve this mysterious problem. However, many results exist and give good approximations of the linear complementarity problem solutions. The major drawback of many existing methods resides in the fact that, for large systems, they require a large number of operations during each iteration; also, they consume large amounts of memory and computation time. This is the reason which drives us to create an algorithm with a finite number of steps to solve this kind of problem with a reduced number of iterations compared to existing methods. In addition, we consider a new class of matrices called the E-matrix.

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