Horizontal Linear Complementarity Problem Research Articles

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.

General double-relaxation two-sweep modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

General double-relaxation two-sweep modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

Convergence Analysis of the Projected SOR Iteration Method for Horizontal Linear Complementarity Problems

Convergence Analysis of the Projected SOR Iteration Method for Horizontal Linear Complementarity Problems

Nonsmooth dynamic modeling and simulation of spatial mechanisms with frictional translational clearance joints

This paper presents a nonsmooth dynamic approach of modeling and computation for spatial mechanism including spatial frictional translational joints (SFTJs) with small clearances. In this research, the dynamic formulation is derived in the Lie group setting, which leads to a coordinate-free and compact formulation. In the following, the normal contact interaction between the slider and guide is described by the complementarity relations between the constraint force in the normal direction. The tangent contact interaction in the SFTJ is characterized by the Coulomb’s friction law in the type of a set-valued map. Based on the horizontal linear complementarity problem (HLCP), the non-smooth dynamics can be established and calculated by combining the Lemke’s algorithm and the RK-MK time integration algorithm. Finally, A spatial crank-slider mechanism with SFTJs is shown as a numerical case. The simulation results demonstrate the correctness and effectiveness of the proposed method which can capture the nonsmooth dynamical behavior of the system.

On finiteness of the solution set of extended horizontal linear complementarity problem

In this paper, we introduce the concept of column nondegenerate-W property to the set of matrices, and we show that when the involved set of matrices have the column nondegenerate-W property is equivalent to the Extended Horizontal Linear Complementarity Problem have the finiteness property of the solution set.

Convergence of a smoothing algorithm for horizontal linear complementarity problem with a nonmonotone line search

In this paper, we present a nonmonotone smoothing Newton type algorithm for solving the horizontal linear complementarity problem. We show that the proposed algorithm is globally convergent under an assumption that has the column -property. Such an assumption is weaker than those required by most other smoothing Newton algorithms. The preliminary numerical results are reported.

The solvability of weighted complementarity problems and a smoothing Newton algorithm under the local error bound

The weighted Complementarity Problem (wCP) can be used for modelling a larger class of problems from science and engineering. In this paper, we first extend the solvability of the Complementarity Problems (CP) obtained by Yoshise [SIAM J. Optim. 17, 1129–1153 (2006)] to the wCP. Especially, we prove that the solution set of the wCP is bounded. We then propose a smoothing Newton algorithm to solve the wCP and prove that it is globally convergent. Moreover, we establish the local quadratic convergence of the proposed algorithm under the local error bound condition which is weaker than the nonsingularity condition used in previous smoothing Newton-type algorithms. Further, we apply the proposed algorithm to solve the weighted horizontal linear complementarity problem and report some numerical results.

The nonlinear lopsided PSS-like and HSS-like modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

The nonlinear lopsided PSS-like and HSS-like modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

Interior-point algorithm for symmetric cone horizontal linear complementarity problems based on a new class of algebraically equivalent transformations

AbstractWe generalize a primal-dual interior-point algorithm (IPA) proposed recently in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) to $$P_*(\kappa )$$ P ∗ ( κ ) -horizontal linear complementarity problems (LCPs) over Cartesian product of symmetric cones. The algorithm is based on the algebraic equivalent transformation (AET) technique with a new class of AET functions. The new class is a modification of the class of AET functions proposed in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) where only two conditions are used as opposed to three used in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022). Furthermore, the algorithm is a feasible algorithm that uses full Nesterov-Todd steps, hence, no calculation of step-size is necessary. Nevertheless, we prove that the proposed IPA has the iteration bound that matches the best known iteration bound for IPAs solving these types of problems.

An efficient multi parametric kernel function for large and small-update methods interior point algorithm for P*(κ)-horizontal linear complementarity problem

In this paper, we propose the first efficient multi parametric kernel function with logarithmic barrier term. A class of polynomial interior-point algorithms for P*(κ)-horizontal linear complementarity problem based on this kernel function, with parameters pi > 0 for all i ∈ 1, 2, , m, are presented. Then by using some simple analysis tools, we present a primal-dual interior point method (IPM) for P*(κ)-horizontal linear complementarity problems based on this kernel function. At the same time, we derive the complexity bounds small and large-update methods, respectively. In particular, if we take many different values of the parameters, we obtain the best known iteration bounds for the algorithms with large- and small-update methods are derived, namely, O((1 + 2κ)√n(log n)log n/ϵ) and O((1 + 2κ)√n log n/ϵ) respectively. We illustrate the performance of the proposed kernel function by some numerical results that are derived by applying our algorithm.

An Improved Convergence Condition of the MMS Iteration Method for Horizontal LCP of H+-Matrices

In this paper, inspired by the previous work in (Appl. Math. Comput., 369 (2020) 124890), we focus on the convergence condition of the modulus-based matrix splitting (MMS) iteration method for solving the horizontal linear complementarity problem (HLCP) with H+-matrices. An improved convergence condition of the MMS iteration method is given to improve the range of its applications, in a way which is better than that in the above published article.

Projected fixed point iterative method for large and sparse horizontal linear complementarity problem

Projected fixed point iterative method for large and sparse horizontal linear complementarity problem

A new search direction of IPM for horizontal linear complementarity problems

This study presents a new search direction for the horizontal linear complementarity problem. A vector-valued function is applied to the system of xy=μe, which defines the central path. Usually, the way to get the equivalent form of the central path is using the square root function. However, in our study, we substitute a new search function formed by a different identity map, which obtains the equivalent shape of the central path using the square root function. We get the new search directions from Newton’s Method. Given this framework, we prove polynomial complexity for the Newton directions. We show that the algorithm’s complexity is O(nlognϵ), which is the same as the best-given algorithms for the horizontal linear complementarity problem.

A full-Newton step feasible interior-point algorithm based on a simple kernel function for $latex P_*(\kappa)$-horizontal linear complementarity problem

A full-Newton step feasible interior-point algorithm based on a simple kernel function for $latex P_*(\kappa)$-horizontal linear complementarity problem

A preconditioned general modulus-based matrix splitting iteration method for solving horizontal linear complementarity problems

A preconditioned general modulus-based matrix splitting iteration method for solving horizontal linear complementarity problems

A two-step parallel iteration method for large sparse horizontal linear complementarity problems

In this work, a two-step modulus-based synchronous multisplitting iteration method is constructed for solving large sparse horizontal linear complementarity problems. Some convergence theorems of the proposed method are presented, which can generalize the convergence results of some existing methods. Numerical tests on parallel computers by OpenACC show that the proposed method is efficient.

New Predictor–Corrector Algorithm for Symmetric Cone Horizontal Linear Complementarity Problems

We propose a new predictor–corrector interior-point algorithm for solving Cartesian symmetric cone horizontal linear complementarity problems, which is not based on a usual barrier function. We generalize the predictor–corrector algorithm introduced in Darvay et al. (SIAM J Optim 30:2628–2658, 2020) to horizontal linear complementarity problems on a Cartesian product of symmetric cones. We apply the algebraically equivalent transformation technique proposed by Darvay (Adv Model Optim 5:51–92, 2003), and we use the difference of the identity and the square root function to determine the new search directions. In each iteration, the proposed algorithm performs one predictor and one corrector step. We prove that the predictor–corrector interior-point algorithm has the same complexity bound as the best known interior-point methods for solving these types of problems. Furthermore, we provide a condition related to the proximity and update parameters for which the introduced predictor-corrector algorithm is well defined.

A Mehrotra Type Predictor-Corrector Interior-Point Method for -HLCP

In this article, a new variant of Mehrotra type-predictor-corrector algorithm is proposed for -horizontal linear complementarity problem. We demonstrate the theoretical efficiency of this algorithm by showing its polynomial complexity. The iteration bound is given by We test the practical efficiency and the validity of our algorithm by running some computational tests.

A relaxation two-sweep modulus-based matrix splitting iteration method for horizontal linear complementarity problems

A relaxation two-sweep modulus-based matrix splitting iteration method for horizontal linear complementarity problems

A sign-based linear method for horizontal linear complementarity problems

A sign-based linear method for horizontal linear complementarity problems