Linear Complementarity Problem Research Articles

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

On the two-stage multisplitting iteration methods for linear complementarity problems

In this work, the two-stage multisplitting iteration methods based on the equivalent modulus equations are analyzed for solving linear complementarity problems. New convergence results are presented where the convergence domains of the parameter matrices are enlarged compared the existing literatures. Furthermore, the improved domains are validated by numerical examples with parallel computations using OpenACC.

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

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

Real option pricing under the regime-switching model with jumps on a finite time horizon

We consider an irreversible investment decision problem on a finite time horizon where an instantaneous cash flow process of a firm follows a regime-switching jump-diffusion (RSJD) model. The value of a project can be derived from a partial integro-differential equation (PIDE) and then we obtain a closed-form solution of the PIDE. It is proved that the value of the project converges to the solution on an infinite time horizon problem as the lifetime of the project tends to infinity. Moreover, the value function of a real option with a finite expiration date can be evaluated by solving a linear complementarity problem (LCP) under the RSJD model. We apply a 2-step backward differentiation formula (BDF2 method) combined with an operator splitting method to solve the LCP and then it has the second-order convergence rate in the discrete ℓ2-norm with respect to the time and spatial variables. The stability of the proposed method to solve the LCP is also proved in the discrete ℓ2-norm. Finally, a variety of numerical experiments are carried out to determine the optimal investment time and the numerical results on the finite time horizon are analyzed.

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

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.

Operator Splitting Method to Solve the Linear Complementarity Problem for Pricing American Option: An Approximation of Error

Operator Splitting Method to Solve the Linear Complementarity Problem for Pricing American Option: An Approximation of Error

On the number of pivots of Dantzig's simplex methods for linear and convex quadratic programs

Refining and extending works by Ye and Kitahara-Mizuno, this paper presents new results on the number of pivots of simplex-type methods for solving linear programs of the Leontief kind, certain linear complementarity problems of the P kind, and nonnegatively constrained convex quadratic programs. Our results contribute to the further understanding of the complexity and efficiency of simplex-type methods for solving these problems. Two applications of the quadratic programming results are presented.

Withdrawal model for fiber motions in the tuft disentanglement

Tuft disentanglement involves the principal actions of opening and carding, which are essential processing in textile technology. In this article, tuft disentanglement is modeled as fiber withdrawal considering the fiber interactions within the tuft. Based on the reasonable tuft model, the kinematic constraints were constructed in the disentanglement considering the carding essence and fiber interactions. Combined with the dynamic equation, the state of the fibers can be solved as a linear complementarity problem due to the complementarity in the fiber interactions. The withdrawal forces verified the simulations, which capture the qualitative features of experiments. Also, the effect of the withdrawal parameters on disentanglement was studied. The simulation results showed that the withdrawal forces increased with increasing withdrawal velocities and decreasing gauge lengths. The simulation results are consistent with the experiments. The proposed computational framework can be extended for predicting the mechanical behaviors of other random fibrous materials involving fiber interactions.

A Full-Newton Step Feasible Interior-Point Algorithm for the Special Weighted Linear Complementarity Problems Based on a Kernel Function

In this paper, a new full-Newton step primal-dual interior-point algorithm for solving the special weighted linear complementarity problem is designed and analyzed. The algorithm employs a kernel function with a linear growth term to derive the search direction, and by introducing new technical results and selecting suitable parameters, we prove that the iteration bound of the algorithm is as good as best-known polynomial complexity of interior-point methods. Furthermore, numerical results illustrate the efficiency of the proposed method.

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.

A novel approach to establishing uniqueness of solutions in linear complementarity problems

Linear complementarity problems are fundamental in various mathematical fields, and it is widely recognized that these problems have a unique solution if and only if the matrix related to them is a P-matrix. However, determining whether a matrix is a P-matrix can be extremely challenging or even impossible, especially for large matrices. In this paper, we present a method for establishing the existence and uniqueness of the solution to linear complementarity problems without the need for P-matrix verification.

A relaxed two-step modulus-based matrix synchronous multisplitting iteration method for linear complementarity problems

A relaxed two-step modulus-based matrix synchronous multisplitting iteration method for linear complementarity problems

Approximate Message Passing for sparse matrices with application to the equilibria of large ecological Lotka–Volterra systems

This paper is divided into two parts. The first part is devoted to the study of a class of Approximate Message Passing (AMP) algorithms which are widely used in the fields of statistical physics, machine learning, or communication theory. The AMP algorithms studied in this part are those where the measurement matrix has independent elements, up to the symmetry constraint when this matrix is symmetric, with a variance profile that can be sparse. The AMP problem is solved by adapting the approach of Bayati, Lelarge, and Montanari (2015) to this matrix model.The Lotka–Volterra (LV) model is the standard model for studying the dynamical behavior of large dimensional ecological food chains. The second part of this paper is focused on the study of the statistical distribution of the globally stable equilibrium vector of a LV system in the situation where the random symmetric interaction matrix among the living species is sparse, and in the regime of large dimensions. This equilibrium vector is the solution of a Linear Complementarity Problem, which distribution is shown to be characterized through the AMP approach developed in the first part. In the large dimensional regime, this distribution is close to a mixture of a large number of truncated Gaussians.

Algorithm for calculating the problem of unilateral frictional contact with an increscent external load parameter

The subject of the study is the contact interaction of deformable elements of linear complementarity problem (LCP). To solve the linear complementarity problem, the Lemke method with the introduction of an increasing parameter of external loading is used. The proposed approach solves the degenerated matrix in a finite number of steps, while the dimensionality of the problem is limited to the area of contact. To solve the problem, the initial table of the Lemke method is generated using the contact matrix of stiffness and the contact load vector. The unknowns in the problem are mutual displacements and interaction forces of contacting pairs of points of deformable solids. The proposed approach makes it possible to evaluate the change in working schemes as the parameter of external load increases. The features of the proposed formulation of the problem are shown, the criteria for stopping the stepwise process of solving such problems are considered. Model examples for the proposed algorithm are given. The algorithm has shown its efficiency in application, including for complex model problems. Recommendations on the use of the proposed approach are given.

A full-Newton step interior-point algorithm for the special weighted linear complementarity problem based on positive-asymptotic kernel function

A full-Newton step interior-point algorithm for the special weighted linear complementarity problem based on positive-asymptotic kernel function

Projection and Contraction Method for Pricing American Bond Options

In this paper, an effective numerical method is proposed for a linear complementarity problem (LCP) arising in the valuation of American bond options under the Cox–Ingersoll–Ross (CIR) model. Firstly, a variable substitution is used to simplify the linear complementary model. Secondly, the finite difference method is adopted to discretize the simplified model, and an equivalent variational form is obtained. Based on the positive definiteness of the discretized matrix, a projection and contraction method (PCM) is adopted for the resulting discretized variational problem. Finally, numerical experiments highlight the effectiveness and performance of the proposed algorithm.

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).

Efficient pricing of options in jump–diffusion models: Novel implicit–explicit methods for numerical valuation

This paper presents novel implicit–explicit Runge–Kutta type methods for numerically simulating partial integro-differential equations that arise when pricing options under jump–diffusion models. These methods offer an alternative approach that avoids the need for numerical or analytical inversion of the coefficient matrix. The pricing of European options is formulated as a partial integro-differential equation, while the pricing of American options are treated as a linear complementarity problem. The developed implicit–explicit Runge–Kutta type method is combined with an operator splitting technique to efficiently solve the linear complementarity problem. Stability and convergence analysis of the proposed methods are established using discrete ℓ2-norm. To validate their efficiency and accuracy, the methods are applied to pricing European and American options under Merton’s and Kou’s models, and the computed results are compared with those reported in the literature.

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.