Abstract
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 the L‐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.
Highlights
Complementarity theory introduced and studied by Lemke 1 and Cottle and Dantzig 2 has enjoyed a vigorous growth for the last fifty years
If K is the positive cone in Rn, x ∈ K is a solution of absolute value variational inequality 1.2 if and only if x ∈ K solves the absolute value complementarity problem 1.1
The convergence of the generalized AOR method is guaranteed for L-matrices only but it is possible to solve different types of systems
Summary
Complementarity theory introduced and studied by Lemke 1 and Cottle and Dantzig 2 has enjoyed a vigorous growth for the last fifty years. It is well known that both the linear and nonlinear programs can be characterized by a class of complementarity problems. The complementarity problems have been generalized and extended to study a wide class of problems, which arise in pure and applied sciences; see 1–24 and the references therein. Important is the variational inequality problem, which was introduced and studied in the early sixties. The theory of variational inequality has been developed to study the fundamental facts on the qualitative behavior of solutions and to provide highly efficient new numerical methods for solving various nonlinear problems. Formulation, numerical results, and other aspects of the variational inequalities, see 13–22
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