Standard Linear Complementarity Problem Research Articles

A Two-steps Fixed-point Method for the Simplicial Cone Constrained Convex Quadratic Optimization

In this paper, we deal with the resolution of the simplicial cone constrained convex quadratic optimization (abbreviated SCQO). It is known that the optimality conditions of SCQO is only a standard linear complementarity problem (LCP). Under a suitable condition, the solution of LCP is equivalent to find the solution of an absolute value equations AVE. For its numerical solution, we propose an efficient two-steps fixed point iterative method for solving the AVE. Moreover, we show that this method converges globally linear to the unique solution of the AVE and which is in turn an optimal solution of SCQO. Some numerical results are reported to demonstrate the efficiency of the proposed algorithm. Keywords: Quadratic programming, simplicial cones, absolute value equations, linear complementarity problem, Picard’s fixed point iterative method.

Structure of the w-solution set of the tensor complementarity problem

Motivated by the study of column competent and column adequate matrices in the standard linear complementarity problems, we define column competent tensor, column adequate tensor and study the underlying properties of these tensors. We establish equivalent conditions for a column adequate tensor within the class of symmetric tensors and Z tensors. We then introduce the -solution set for the tensor complementarity problem and examine the uniqueness and finiteness of the -solution set. At last, we define column competent property and prove that under some additional condition, column competent property gives the finiteness of the -solution set of the corresponding tensor complementarity problem.

Exact computation of an error bound for the balanced linear complementarity problem with unique solution

This paper considers the balanced form of the standard linear complementarity problem with unique solution and provides a more precise expression of an upper error bound discovered by Chen and Xiang and published in 2006. This expression has at least two advantages. It makes possible the exact computation of the error bound factor and it provides a satisfactory upper estimate of that factor in terms of the data bitlength when the data is formed of rational numbers. Along the way, we show that, when any rowwise convex combination of two square matrices is nonsingular, the $$\ell _\infty $$ norm of the inverse of these rowwise convex combinations is maximized by an extreme diagonal matrix.

A Modified Discrete Scheme in the Ausas Cavitation Algorithm

Abstract In order to reduce the dependence of accuracy on the number of grids in the Ausas cavitation algorithm, a modified Ausas algorithm was presented. By modifying the mass-conservative Reynolds equation with the concept of linear complementarity problems (LCPs), the coupling of film thickness h and density ratio θ disappeared. The modified equation achieved a new discrete scheme that ensured a complete second-order-accurate central difference scheme for the full film region, avoiding a hybrid-order-accurate discrete scheme. A journal-bearing case was studied to show the degree of accuracy improvement and the calculation time compared to a standard LCP solver. The results showed that the modified Ausas algorithm made the asymptotic and convergent behavior with the increase of nodes disappear and allowed for the use of coarse meshes to obtain sufficient accuracy. The calculation time of the modified Ausas algorithm is shorter than that of the LCP solver (Lemke’s pivoting algorithm) for middle- and large-scale problems.

On unique solvability and Picard’s iterative method for absolute value equations

In this paper, we deal with unique solvability and numerical solution ofabsolute value equations (AVE),Ax−B|x|=b, (A,B∈Rn×n,b∈Rn).Under some weaker conditions, a simple proof is given for unique solvabilityof AVE. Furthermore, we demonstrate with an example that these results arereliable to detect unique solvability of AVE. These results are also extendedto unique solvability of standard and horizontal linear complementarity prob-lems. Finally, we suggest a Picard iterative method to compute an approx-imated solution of some uniquely solvable AVE problems where its globallylinear convergence is guaranteed via one of our weaker sufficient condition.

Modulus-based matrix splitting methods for a class of horizontal nonlinear complementarity problems

In this paper, we generalize modulus-based matrix splitting methods to a class of horizontal nonlinear complementarity problems (HNCPs). First, we write the HNCP as an implicit fixed-point equation and we introduce the proposed solution procedures. We then prove the convergence of the methods under some assumptions. We also comment on how the proposed methods and convergence theorems generalize existing results on (standard) linear and nonlinear complementarity problems and on horizontal linear complementarity problems. Finally, numerical experiments are solved to demonstrate the efficiency of the procedures. In this context, the effects of the splitting, of the dimension of the matrices, and of the nonlinear term of the problem are analyzed.

A symplectic pseudospectral method for constrained time-delayed optimal control problems and its application to biological control problems

ABSTRACT Time-delays are often involved in modelling biological processes, which arise more difficulties for the control task. In fact, once the control goal is given, a constrained optimal control problem can be formulated to describe the biological control mission. However, it’s almost impossible to obtain analytical solutions for such complicated problems, especially for those with high nonlinearity and complex constraints. In this paper, a symplectic pseudospectral method for solving nonlinear optimal control problems of constrained stated-delayed system is developed. The original nonlinear problem is transformed into a series of linear-quadratic optimal control problems by the successive convexification technique at the first implementation. By applying the local Legendre-Gauss-Lobatto pseudospectral method and the parametric variational principle, each of the transformed linear-quadratic problems is converted into a standard linear complementarity problem which can be solved easily by the Lemke’s method. The cost functional, system dynamics and constraints can be generally nonlinear functions of state, control and time variables. and three kinds of inequality constraints, i.e. pure-state, pure-control and mixed state-control, can be treated under a uniform framework. The developed symplectic pseudospectral method is finally validated by an optimal harvesting problem and an optimal vaccination problem.

A modulus-based nonsmooth Newton’s method for solving horizontal linear complementarity problems

In this paper, we introduce and analyze a modulus-based nonsmooth Newton’s method for solving horizontal linear complementarity problems. In particular, we propose both a standard and a parametrized formulation of the method, proving the local convergence of the approaches. The proposed procedures generalize the existing modulus-based nonsmooth Newton’s method for standard linear complementarity problems. Then, we present an implementation of the methods, analyzing also the coupling with a modulus-based matrix splitting iteration. Finally, numerical experiments demonstrate the effectiveness of the proposed approaches in several situations.

A Symplectic Instantaneous Optimal Control for Robot Trajectory Tracking With Differential-Algebraic Equation Models

Robot trajectory tracking control based on differential-algebraic equation (DAE) models is still a thorny issue, because the DAEs of such systems are inherently complex and unstable, such as the high-index problem. In this paper, a symplectic instantaneous optimal control (IOC) method for robot trajectory tracking with input saturation based on controlled DAEs is proposed. Based on the discrete variational principle and the canonical transformation, a symplectic discretization form for the controlled DAEs is first constructed. Then, the continuous trajectory tracking problem is approximated for a series of IOC problems at every time step, and the linear complementarity problem (LCP) can be derived for solving the IOC problems. Finally, the control inputs can be obtained by solving the corresponding standard LCP. The proposed method provides a unified framework for solving the trajectory tracking control problems of robot multibody dynamic systems. Numerical simulations and virtual experiments are conducted to verify the robustness and the efficiency of the proposed method, i.e., the input saturation constraints are satisfied at the discrete time points, and high accuracy tracking control results can be obtained at low computational cost.

Contact Analysis of Functionally Graded Materials Using Smoothed Finite Element Methods

In the paper, the smoothed finite element method (S-FEM) based on linear triangular elements is used to solve 2D solid contact problems for functionally graded materials. Both conforming and nonconforming contacts algorithms are developed using modified Coulomb friction contact models including tangential strength and normal adhesion. Based on the smoothed Galerkin weak form, the system stiffness matrices are created using the formulation procedures of node-based S-FEM (NS-FEM) and edge-based S-FEM (ES-FEM), and the contact interface equations are discretized by contact point-pairs. Then these discretized system equations are converted into a form of linear complementarity problems (LCPs), which can be further solved efficiently using the Lemke method. The singular value decomposition method is used to deal with the singularity of the stiffness matrices in the procedure constructing the standard LCP, which can greatly improve the stability and accuracy of the numerical results. Numerical examples are presented to investigate the effects of the various parameters of functionally graded materials and comparisons have been made with reference solutions and the standard FEM. The numerical results demonstrate that the strain energy solutions of ES-FEM have higher convergence rate and accuracy compared with that of NS-FEM and FEM for functionally graded materials through the present contact analysis approach.

A homogeneous model for monotone mixed horizontal linear complementarity problems

We propose a homogeneous model for the class of mixed horizontal linear complementarity problems. The proposed homogeneous model is always solvable and provides the solution of the original problem if it exists, or a certificate of infeasibility otherwise. Our formulation preserves the sparsity of the original formulation and does not reduce to the homogeneous model of the equivalent standard linear complementarity problem. We study the properties of the model and show that interior-point methods can be used efficiently for the numerical solutions of the homogeneous problem. Numerical experiments show convincingly that it is more efficient to use the proposed homogeneous model for the mixed horizontal linear complementarity problem than to use known homogeneous models for the equivalent standard linear complementarity problem.

Stabilizing constrained chaotic system using a symplectic psuedospectral method

The problem of controlling chaotic systems has drawn much attention in the last two decades. However, the controlled system may be subjected to complicated constraints and few researches on controlling chaos take constraints into consideration. Therefore, the stabilization of constrained chaotic system is solved under the frame of nonlinear optimal control in this paper. A symplectic pseudospectral method based on qusilinearizaiton techniques and the parametric variational principle is developed to solve constrained nonlinear optimal control problems with arbitrary Lagrange-type cost functional. At the beginning of the proposed method, the original nonlinear optimal control problem is converted into a series of linear-quadratic constrained optimal control problems. Then each of the converted linear quadratic problems is transformed into a standard linear complementarity problem. The proposed method is successfully applied to stabilizing constrained chaotic systems around an unstable equilibrium point or an unstable periodic orbit. Numerical simulations demonstrate that the developed method is effective and efficient, and constraints are strictly satisfied.

A linear complementarity formulation of meshfree method for elastoplastic analysis of gradient-dependent plasticity

This work presents a linear complementarity formulation for elastoplastic analysis of the gradient-dependent plasticity including large deformation problems. The formulation is based on the meshfree smoothed radial point interpolation method, where the parametric variational principle (PVP) is used in the form of linear complementarity and the gradient-dependent plasticity is represented by the linearization of yield criterion. The yield stress is linearly evolved through equivalent plastic strain as well as its Laplacian (namely second gradient). The global discretized system equations are transformed into a standard linear complementarity problem (LCP), which can be solved readily using the Lemke method. The proposed approach is capable of simulating material hardening/softening and strain localization. An extensive numerical study is performed to validate the proposed method and to investigate the effects of various parameters. The numerical results demonstrate that the proposed approach is accurate and stable for the elastoplastic analysis of 2D solids with gradient-dependent plasticity on strain localization.

A Complementarity Problem–Based Solution Procedure for 2D Steady-State Rolling Contacts with Dry Friction

ABSTRACTThe problem of steady-state rolling contact between two cylinders with dry friction was formulated into standard linear complementarity problems (LCPs) using the explicit physical definition. For normal contacts, the complementarity variables are the normal pressure and the gap. For the tangential contact, the traction distribution and relative slip are the variables obtained by solving the LCP. The frictional behavior is assumed to be governed by the Coulomb friction law, and LCP formulations of both similar elastic (Carter problem) and dissimilar elastic rolling contacts are presented in this work. Good agreement was found between the current LCP approach and publicly available software for both the rolling contact of similar elastic and dissimilar elastic cylinders. Moreover, the surface roughness was taken into account in this article by the verified approach. The results show the initial slope of the traction-relative creepage curve decreases as the surface roughness increases.

A 3D PVP co-rotational formulation for large-displacement and small-strain analysis of bi-modulus materials

An efficient computational method is developed for large-displacement and small-strain analysis of 3D bi-modulus materials which are often found in civil, composite and biological engineering. Based on the parametric variational principle (PVP), a unified constitutive equation of 3D bi-modulus materials is proposed to deal with the problem of numerical instability. The small-strain bi-modulus problem is transformed into a standard linear complementarity problem that can be solved easily by the classic Lemke׳s algorithm. By using the co-rotational approach, the local PVP formulation is combined with an existing co-rotational formulation and a new tangent stiffness matrix, including a parametric variable, is derived for geometrically nonlinear analysis. Traditional stress iteration is not required for calculation of the nodal internal force when the Newton–Raphson scheme or Arc-length method is employed to solve the material and geometric nonlinear problem. Convergence of the proposed algorithm is improved greatly in contrast to the traditionally iterative solution. Also, the proposed algorithm is used to simulate a unilateral contact behavior of staggered bio-composites and the effective elasticity modulus of composites is determined accurately.

A stabilized complementarity formulation for nonlinear analysis of 3D bimodular materials

Bi-modulus materials with different mechanical responses in tension and compression are often found in civil, composite, and biological engineering. Numerical analysis of bimodular materials is strongly nonlinear and convergence is usually a problem for traditional iterative schemes. This paper aims to develop a stabilized computational method for nonlinear analysis of 3D bimodular materials. Based on the parametric variational principle, a unified constitutive equation of 3D bimodular materials is proposed, which allows the eight principal stress states to be indicated by three parametric variables introduced in the principal stress directions. The original problem is transformed into a standard linear complementarity problem (LCP) by the parametric virtual work principle and a quadratic programming algorithm is developed by solving the LCP with the classic Lemke’s algorithm. Update of elasticity and stiffness matrices is avoided and, thus, the proposed algorithm shows an excellent convergence behavior compared with traditional iterative schemes. Numerical examples show that the proposed method is valid and can accurately analyze mechanical responses of 3D bimodular materials. Also, stability of the algorithm is greatly improved.

A linear complementarity method for dynamic analysis of bridges under moving vehicles considering separation and surface roughness

In the present paper, a linear complementarity method for a vehicle-bridge dynamic system considering separation and random roughness is established. By introducing the linear complementarity relationship between the relative displacement of the wheels and the bridge at the contact points, the dynamic interaction problem of the vehicle-bridge coupled system is transformed into a standard linear complementarity problem, and two models with different connection relations between the wheels and the bridge are proposed. The presented models characterize the system with one unified formulation whether the wheels separate from the bridge or not, and the conventional trial-and-error iterative process in numerical simulation is avoided. In the numerical examples, the proposed method is verified by comparing it with the conventional method, and it is found that the velocity, the vehicle to bridge mass ratio and the road roughness have a significant influence on separation. By considering a vehicle model of three rigid bodies with four wheels and the randomness of the rail roughness in train–track–bridge system, the possibility of separation and the expectation of the maximum separation distance at different velocities are studied. The results show that it is very useful to carry out a stochastic analysis of the system and consider the influence of separation in vehicle and bridge design.

SOLVING STRONGLY MONOTONE LINEAR COMPLEMENTARITY PROBLEMS

Given a linear transformation L on a finite dimensional real inner product space V to itself and an element q ∈ V we consider the general linear complementarity problem LCP (L, K, q) on a proper cone K ⊆ V. We observe that the iterates generated by any closed algorithmic map will converge to a solution for LCP (L, K, q), whenever L is strongly monotone. Lipschitz constants of L is vital in establishing the above said convergence. Hence we compute the Lipschitz constants for certain classes of Lyapunov, Stein and double-sided multiplicative transformations in the setting of semidefinite linear complementarity problems. We give a numerical illustration of a closed algorithmic map in the setting of a standard linear complementarity problem. On account of the difficulties in numerically implementing such algorithms for general linear complementarity problems, we give an alternative algorithm for computing the solution for a special class of strongly monotone semidefinite linear complementarity problems along with a numerical example.

A contact analysis approach based on linear complementarity formulation using smoothed finite element methods

Based on the subdomain parametric variational principle (SPVP), a contact analysis approach is formulated in the incremental form for 2D solid mechanics problems discretized using only triangular elements. The present approach is implemented for the newly developed node-based smoothed finite element method (NS-FEM), the edge-based smoothed finite element method (ES-FEM) as well as standard FEM models. In the approach, the contact interface equations are discretized by contact point-pairs using a modified Coulomb frictional contact model. For strictly imposing the contact constraints, the global discretized system equations are transformed into a standard linear complementarity problem (LCP), which can be readily solved using the Lemke method. This approach can simulate different contact behaviors including bonding/debonding, contacting/departing, and sticking/slipping. An intensive numerical study is conducted to investigate the effects of various parameters and validate the proposed method. The numerical results have demonstrated the validity and efficiency of the present contact analysis approach as well as the good performance of the ES-FEM method, which provides solutions of about 10 times better accuracy and higher convergence rate than the FEM and NS-FEM methods. The results also indicate that the NS-FEM provides upper-bound solutions in energy norm, relative to the fact that FEM provides lower-bound solutions.

An efficient computational method for vibration analysis of unsymmetric piecewise-linear dynamical systems with multiple degrees of freedom

This paper presents an efficient computational method for determining vibrational responses of piecewise-linear dynamical systems with multiple degrees of freedom (MDOF) and an arbitrary number of gap-activated springs. A time-domain solution is obtained using the Bozzak–Newmark numerical integration scheme. At each time step, an auxiliary displacement vector, complementary to the contact force vector, is introduced. With the help of a simple transformation, the vibration problem is reduced to a standard linear complementarity problem (LCP) for which an accurate solution can be obtained. Responses of an SDOF system with a gap-activated spring and a 3-DOF system with three gap-activated springs under harmonic excitations are obtained using the proposed method, and compared with the results in the literature. Good agreement is observed. The proposed method has also been successfully applied to a piecewise linear dynamical system with 1000 DOF’s and 1000 gap-activated springs under harmonic excitations.