Merit functions for absolute value variational inequalities

<abstract><p>This article deals with a class of variational inequalities known as absolute value variational inequalities. Some new merit functions for the absolute value variational inequalities are established. Using these merit functions, we derive the error bounds for absolute value variational inequalities. Since absolute value variational inequalities contain variational inequalities, absolute value complementarity problem and system of absolute value equations as special cases, the findings presented here recapture various known results in the related domains. The conclusions of this paper are more comprehensive and may provoke futuristic research.</p></abstract>

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The Picard-HSS-SOR iteration method for absolute value equations

Journal of Inequalities and Applications ◽

10.1186/s13660-020-02525-3 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Lin Zheng

Keyword(s):

Iteration Method ◽

Numerical Experiments ◽

Absolute Value Equation ◽

Absolute Value Equations ◽

Absolute Value ◽

Convergence Results ◽

Hss Iteration ◽

The Absolute ◽

Hss Iteration Method

AbstractIn this paper, we present the Picard-HSS-SOR iteration method for finding the solution of the absolute value equation (AVE), which is more efficient than the Picard-HSS iteration method for AVE. The convergence results of the Picard-HSS-SOR iteration method are proved under certain assumptions imposed on the involved parameter. Numerical experiments demonstrate that the Picard-HSS-SOR iteration method for solving absolute value equations is feasible and effective.

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Generalized AOR Method for Solving Absolute Complementarity Problems

Journal of Applied Mathematics ◽

10.1155/2012/743861 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14 ◽

Cited By ~ 5

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.

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Merit functions and error bounds for generalized variational inequalities

Journal of Mathematical Analysis and Applications ◽

10.1016/s0022-247x(02)00554-1 ◽

2003 ◽

Vol 287 (2) ◽

pp. 405-414 ◽

Cited By ~ 29

Author(s):

M.V. Solodov

Keyword(s):

Variational Inequalities ◽

Error Bounds ◽

Merit Functions ◽

Generalized Variational Inequalities

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Conditions for Error Bounds and Bounded Level Sets of Some Merit Functions for the Second-Order Cone Complementarity Problem

Journal of Optimization Theory and Applications ◽

10.1007/s10957-007-9279-9 ◽

2007 ◽

Vol 135 (3) ◽

pp. 459-473 ◽

Cited By ~ 6

Author(s):

J.-S. Chen

Keyword(s):

Error Bounds ◽

Complementarity Problem ◽

Level Sets ◽

Second Order ◽

Merit Functions ◽

Second Order Cone ◽

Bounded Level Sets

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Error bounds for general mixed quasivariational inequalities

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.1085 ◽

2005 ◽

Vol 2005 (7) ◽

pp. 1085-1100

Author(s):

Muhammad Aslam Noor ◽

Khalida Inayat Noor

Keyword(s):

Fixed Point ◽

Error Bounds ◽

Fixed Point Problems ◽

Merit Functions ◽

Quasivariational Inequalities ◽

Equivalent Formulation ◽

Mild Conditions ◽

Special Cases

It is well known that mixed quasivariational inequalities are equivalent to the implicit fixed-point problems. We use this alternative equivalent formulation to suggest and consider some merit functions for general mixed quasivariational inequalities. We use these merit functions to obtain error bounds for the solution under some mild conditions. Some special cases are also discussed.

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A Preconditioned AOR Iterative Method for the Absolute Value Equations

International Journal of Computational Methods ◽

10.1142/s0219876217500165 ◽

2017 ◽

Vol 14 (02) ◽

pp. 1750016 ◽

Cited By ~ 8

Author(s):

Cui-Xia Li

Keyword(s):

Iterative Method ◽

Comparison Theorems ◽

Absolute Value Equations ◽

Absolute Value ◽

Comparison Results ◽

Preconditioning Technique ◽

The Matrix ◽

The Absolute ◽

Theoretical Results ◽

Better Than

In this paper, coupled with preconditioning technique, a preconditioned accelerated over relaxation (PAOR) iterative method for solving the absolute value equations (AVEs) is presented. Some comparison theorems are given when the matrix of the linear term is an irreducible [Formula: see text]-matrix. Comparison results show that the convergence rate of the PAOR iterative method is better than that of the accelerated over relaxation (AOR) iterative method whenever both are convergent. Numerical experiments are provided in order to confirm the theoretical results studied in this paper.

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