Vertical Linear Complementarity Problem Research Articles

Global Error Bounds for the Extended Vertical LCP of DZ matrices and DZ-B maxtrices

Abstract The linear complementarity problem is a mathematical problem with important theoretical value and wide practical applications. Research on linear complementarity problems mainly focuses on algorithm improvement and innovation; Combining with fields such as machine learning and artificial intelligence to expand the application scope of linear complementarity problems; Further research on the properties of linear complementarity problems, the existence and uniqueness of solutions, and other theoretical issues. In this paper, by virtue of the properties of DZ-matrices and DZ-B matrices, error bound estimation formulas for the extended vertical linear complementarity problem are given respectively for block matrices composed of DZ-matrices and DZ-B matrices under the condition of satisfying the row W-property. At the same time, the feasibility and superiority of the results are illustrated by numerical examples.

The weighted vertical linear complementarity problem on Euclidean Jordan algebra

The weighted vertical linear complementarity problem on Euclidean Jordan algebra

On the modulus-based methods without auxiliary variable for vertical linear complementarity problems

On the modulus-based methods without auxiliary variable for vertical linear complementarity problems

The horizontal tensor complementarity problem

This article explores a new type of nonlinear complementarity problem, namely the horizontal tensor complementarity problem (HTCP), which is a natural extension of the horizontal linear complementarity problem studied in Sznajder [Degree-theoretic analysis of the vertical and horizontal linear complementarity problems [Ph.D. Thesis]. University of Maryland Baltimore County; 1994]. We extend the concepts of R 0 , R and P pairs from a pair of linear transformations given in Chi et al. [The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra. J Global Optim. 2019;73:153–169] to a pair of tensors. When a given pair of tensors has these properties, we use degree-theoretic tools to discuss the existence and boundedness of solutions to the HTCP. Finally, we study a uniqueness result of the solution of the HTCP.

The projected-type method for the extended vertical linear complementarity problem revisited

The projected-type method for the extended vertical linear complementarity problem revisited

A relaxation two-step parallel modulus method without auxiliary variable for solving large sparse vertical linear complementarity problems

A relaxation two-step parallel modulus method without auxiliary variable for solving large sparse vertical linear complementarity problems

Generalized modulus-based matrix splitting algorithm with Anderson acceleration strategy for vertical linear complementarity problems

The vertical linear complementarity problem (VLCP) with an arbitrary number l of matrices is related to many practical problems, of which the state-of-the-art modulus-based matrix splitting (MMS) method is proved to be an efficient solver. Enlightened by the Anderson acceleration which is a well-established and simple technique for speeding up fixed point iteration solvers with countless applications, we propose an Anderson accelerated generalized modulus-based matrix splitting (AA+GMMS) method for solving the VLCP. We particularly analyze the AA+GMMS method for the problem with l=2 and then generalize the method to any l. More importantly, the convergence theorems and theoretical optimal parameters of the MMS, AA+GMMS methods with any l are obtained in the positive definite case. Eventually, numerical experiments are given to demonstrate the effectiveness of the AA+GMMS method which significantly accelerates the original MMS method. In particular, we explore the parameters involved in the AA+GMMS method, and they have a small extent of impact on the suggested method, reinforcing that the AA+GMMS method is highly efficient.

Global error bounds for the extended vertical LCP of $ \sum-SDD $ matrices

<p>An error bound for the solution of the $ \sum-SDD $ matrix extended vertical linear complementarity problem is given when the $ \sum-SDD $ matrix satisfies the row W-property. It is shown by the illustrative example that the new bound is better than those in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> in some cases.</p>

A two-step new modulus-based matrix splitting method for vertical linear complementarity problem

In this paper, for solving the vertical linear complementarity problem (VLCP) effectively, a two-step new modulus-based matrix splitting (TNMMS) iteration method is introduced. Its convergence properties are discussed in depth. The efficiency of the proposed method is confirmed by some numerical experiments from the 2-D boundary problem. Numerical results show that the proposed method is superior to some existing methods, such as the two modulus-based matrix splitting (TMMS) iteration method (Numer. Algor., 2011, 57: 83–99) and the new modulus-based matrix splitting (NMMS) iteration method (Optim., 2023, 72: 2499–2516).

A Two-Step Modulus-Based Matrix Splitting Iteration Method Without Auxiliary Variables for Solving Vertical Linear Complementarity Problems

A Two-Step Modulus-Based Matrix Splitting Iteration Method Without Auxiliary Variables for Solving Vertical Linear Complementarity Problems

Modulus-based synchronous multisplitting iteration methods without auxiliary variable for solving vertical linear complementarity problems

In this work, by applying the synchronous multisplitting technique to the non-auxiliary variable modulus equation of the vertical linear complementarity problems, a new parallel method is constructed, which can generalize the existing modulus-based matrix splitting iteration method. The convergence conditions of the proposed method are discussed in the cases of more than two system matrices, and the existing results are improved. Numerical examples under OpenACC framework show that the proposed method can solve the large sparse vertical linear complementarity problems efficiently with high parallel efficiency.

Relaxation modulus-based matrix splitting iteration method for vertical linear complementarity problem

For solving vertical linear complementarity problems (VLCPs), we propose a relaxation modulus-based matrix splitting iteration method and a two-step relaxation modulus-based matrix splitting iteration method. When the system matrices are H+-matrices, we analyze the convergence theory of the proposed methods. Numerical experiments show that relaxation modulus-based matrix splitting iteration method and the two-step relaxation modulus-based matrix splitting iteration method are promising and have great performance in higher dimension.

Convergence analysis of projected SOR iteration method for a class of vertical linear complementarity problems

Convergence analysis of projected SOR iteration method for a class of vertical linear complementarity problems

Global error bounds for the extended vertical linear complementarity problems of CKV-type matrices and CKV-type B-matrices

Global error bounds for the extended vertical linear complementarity problems of CKV-type matrices and CKV-type B-matrices

The Perturbation Bound of the Extended Vertical Linear Complementarity Problem

In this paper, we discuss the perturbation analysis of the extended vertical linear complementarity problem (EVLCP). Under the assumption of the row $${\mathcal {W}}$$ -property, we derive several absolute and relative perturbation bounds of EVLCP, which extend some existing results. Several numerical examples are given to show the proposed bounds.

Modulus-based synchronous multisplitting iteration methods for large sparse vertical linear complementarity problems

Modulus-based synchronous multisplitting iteration methods for large sparse vertical linear complementarity problems

New convergence results of the modulus-based methods for vertical linear complementarity problems

In this work, for vertical linear complementarity problems, new convergence results of the modulus-based matrix splitting iteration method are given for the general cases, which can also extend and improve the existing ones proposed by “F. Mezzadri, A modulus-based formulation for the vertical linear complementarity problems, Numer. Algorithms 90 (2022) 1547–1568”.

Global error bounds of the extended vertical linear complementarity problems for Dashnic–Zusmanovich matrices and Dashnic–Zusmanovich-B matrices

Global error bounds of the extended vertical linear complementarity problems for Dashnic–Zusmanovich (DZ) matrices and Dashnic–Zusmanovich-B (DZ-B) matrices are presented, respectively. The obtained error bounds are sharper than those of Zhang et al. (Comput. Optim. Appl. 42(3):335–352, 2009) in some cases. Some numerical examples are given to illustrate the obtained results.

A Two-Step Iteration Method for Vertical Linear Complementarity Problems

In this paper, a two-step iteration method was established which can be viewed as a generalisation of the existing modulus-based methods for vertical linear complementarity problems. The convergence analysis of the proposed method is presented, which can enlarge the convergence domain of the parameter matrix compared to the recent results. Numerical examples show that the proposed method is efficient with the two-step technique and confirm the improvement of the theoretical results.

A new kind of modulus-based matrix splitting methods for vertical linear complementarity problems

A new kind of modulus-based matrix splitting methods is proposed to solve the vertical linear complementarity problems in a direct way. This kind of methods is different from the existing modulus-based formulation which based on an equivalent form of the problem. Convergence of the new methods is proved under certain conditions. Numerical experiments are given to show that the efficiency of the new methods is better than existing ones.