Solution Of Linear Complementarity Problem Research Articles

A particle dynamical system algorithm to find the sparse linear complementary solutions

The Linear Complementarity Problem (LCP) offers a comprehensive modeling framework for addressing a wide range of optimization problems. In many real-world applications, finding an LCP solution with a sparse structure is often necessary. To address this problem, we introduce an innovative global optimization framework named the Particle Dynamical System Algorithm (PDSA), which consists of two components. The first component is a dynamical system (DS) inspired by the Absolute Value Equation (AVE), proven to have equilibria corresponding to LCP solutions, with additional relaxing regulators that enhance coverage rate and stability. The second component is an Adaptive Oscillated Particle Swarm Optimization (AOPSO) designed to globally enhance sparsity in LCP solutions, addressing the complexities posed by non-convex and non-smooth regulation models. Within this framework, the DS achieves optimality, while the AOPSO promotes solution sparsity. We compared our proposed DS with relaxing regulators to two classic efficient DSs, fully validating the effectiveness of our approach and underscoring the significant role of the introduced relaxing regulators in improving the convergence rate. Our newly developed variant of PSO, AOPSO, was compared with three classic and state-of-the-art variants on fourteen benchmark functions, demonstrating its competitive performance. Finally, we performed experiments on seven test examples and an application in portfolio selection, showing that the proposed PDSA algorithm surpasses other competitors in finding sparse LCP solutions.

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.

Enhancing Autonomous Vehicle Lateral Control: A Linear Complementarity Model-Predictive Control Approach

Model-predictive control (MPC) offers significant advantages in addressing constraint-related challenges and plays a pivotal role in self-driving car technology. Its primary goal is to achieve precise trajectory tracking while prioritizing vehicle stability and safety. However, real-time operations often face challenges related to computational demands and low computational efficiency. To address these challenges, this paper introduces a novel lateral control algorithm for self-driving vehicles, which utilizes the linear complementarity problem (LCP) instead of the conventional quadratic programming (QP) method as the MPC optimization solution. This innovative approach incorporates the electric steering system into the vehicle dynamics model, allowing for precise torque regulation of the steering motor and enhancing control accuracy. The MPC algorithm adopts the LCP solution method to calculate control signals based on the vehicle’s state, ensuring both rapid and stable vehicle control. Simulation results demonstrate that the proposed MPC algorithm, utilizing the LCP solution method, effectively addresses efficiency issues in the lateral motion of self-driving cars. This leads to improvements in both driving stability and real-time performance. Overall, this innovative approach lays a solid foundation for the practical implementation of self-driving cars.

Bounded homotopy path approach to the solution of linear complementarity problems

Bounded homotopy path approach to the solution of linear complementarity problems

A path-following interior-point algorithm for monotone LCP based on a modified Newton search direction

In this paper, we propose a short-step feasible full-Newton step path-following interior-point algorithm (IPA) for monotone linear complementarity problems (LCPs). The proposed IPA uses the technique of algebraic equivalent transformation (AET) induced by an univariate function to transform the centering equations which defines the central-path. By applying Newton’s method to the modified system of the central-path of LCP, a new Newton search direction is obtained. Under new appropriate defaults of the thresholdτwhich defines the size of the neighborhood of the central-path and ofθwhich determines the decrease in the barrier parameter, we prove that the IPA is well-defined and converges locally quadratically to a solution of the monotone LCPs. Moreover, we derive its iteration bound, namely,O(√nlogn\ϵ) which coincides with the best-known iteration bound for such algorithms. Finally, some numerical results are presented to show its efficiency.

A modified Lemke Algorithm for dynamic rigid plastic response of skeletal structures

This paper proposes a modified Lemke algorithm to determine the non-holonomic response of rigid plastic skeletal structures subjected to extreme dynamic loading. The basic formulation for the dynamic rigid-plastic response has a mathematical form of linear complementarity problem (LCP), and, equivalently, a pair of dual quadratic programs (QP). Previous attempts using the Wolfe type LCP solver exhibited very pronounced sensitivity with semidefinite LCPs, which appear in this class of dynamics. Therefore, the current study offers a numerically robust Lemke algorithm, which is considerably adapted to deal effectively with the presence of unrestricted variables and the semidefinite characteristics of structural matrices. In addition, degeneracy in the LCP solution of rigid plastic structures is efficiently handled by using the Lexicographic Minimum Ratio test. To validate the formulation and the algorithm developed, bench tests are conducted, which involve a simply supported beam and a portal frame subjected to a uniformly distributed rectangular pulse loading.

Γ-robust linear complementarity problems

Complementarity problems are often used to compute equilibria made up of specifically coordinated solutions of different optimization problems. Specific examples are game-theoretic settings like the bimatrix game or energy market models like for electricity or natural gas. While optimization under uncertainties is rather well-developed, the field of equilibrium models represented by complementarity problems under uncertainty – especially using the concepts of robust optimization – is still in its infancy. In this paper, we extend the theory of strictly robust linear complementarity problems (LCPs) to Γ-robust settings, where existence of worst-case-hedged equilibria cannot be guaranteed. Thus, we study the minimization of the worst-case gap function of Γ-robust counterparts of LCPs. For box and -norm uncertainty sets we derive tractable convex counterparts for monotone LCPs and study their feasibility as well as the existence and uniqueness of solutions. To this end, we consider uncertainties in the vector and in the matrix defining the LCP. We additionally study so-called ρ-robust solutions, i.e. solutions of relaxed uncertain LCPs. Finally, we illustrate the Γ-robust concept applied to LCPs in the light of the above mentioned classical examples of bimatrix games and market equilibrium modelling.

Mathematical programs with complementarity constraints and a non-Lipschitz objective: optimality and approximation

We consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessary optimality conditions hold at a local minimizer. In this paper, we present necessary optimality conditions and MPCC-tailored qualifications for the non-Lipschitz MPCC. The proposed qualifications are related to the constraints and the non-Lipschitz term, which ensure that local minimizers satisfy these necessary optimality conditions. Moreover, we present an approximation method for solving the non-Lipschitz MPCC and establish its convergence. Finally, we use numerical examples of sparse solutions of linear complementarity problems and the second-best road pricing problem in transportation science to illustrate the effectiveness of our approximation method for solving the non-Lipschitz MPCC.

The general two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems

In this paper, we present a general two-sweep modulus-based iteration method to solve a class of linear complementarity problems. Convergence analysis shows that the general two-sweep modulus-based matrix splitting iteration method will converge to the exact solution of linear complementarity problem under appropriate conditions. Numerical experiments further show that the proposed methods are superior to the existing methods in actual implementation.

An inexact alternating direction method of multipliers for the solution of linear complementarity problems arising from free boundary problems

A large number of free boundary problems can be formulated as linear-complementarity problems. In this paper, we propose an inexact alternating direction method of multipliers for solving linear complementarity problem arising from free boundary problems by using the special structure of these problems. The convergence of our proposed method is proved. Numerical results show that the proposed method is feasible and effective, and it is significantly faster than modified alternating direction implicit algorithm and many other methods, especially when dimension of the problem being solved is large.

Sparse solution of some special optimization problems

Sparse solution is an active topic with a range of applications. This paper demonstrated some sparse solutions of linear complementarity problem (LCP) and absolute value equation (AVE). Possible future work is that how to compute sparse solution of LCP and AVE with the relations between LCP and AVE.

A note on the unique solution of linear complementarity problem

In this note, the unique solution of the linear complementarity problem (LCP) is further discussed. Using the absolute value equations, some new results are obtained to guarantee the unique solution of the LCP for any real vector.

Copositivity tests based on the linear complementarity problem

We present copositivity tests based on new necessary and sufficient conditions which require the solution of linear complementarity problems (LCP). We propose methodologies involving Lemke's method, an enumerative algorithm and a linear mixed-integer programming formulation to solve the required LCPs. Moreover, we discuss a new necessary condition for (strict) copositivity based on solving a linear program, which can be used as a preprocessing step. The algorithms with these three different variants are thoroughly applied to test matrices from the literature and to max-clique instances with matrices of order up to $$496\times 496$$496×496. We compare our procedures with three other copositivity tests from the literature as well as with a general global optimization solver. The numerical results are very promising and equally good and in many cases better than the results reported elsewhere.

A half thresholding projection algorithm for sparse solutions of LCPs

In this paper, we aim to find sparse solutions of the linear complementarity problems (LCPs), which has many applications such as bimatrix games and portfolio selection. Mathematically, the underlying model is NP-hard in general. Thus, an \(\ell _{1/2}\) regularized projection minimization model is proposed for relaxation. A half thresholding projection (HTP) algorithm is then designed for this regularization model, and the convergence of HTP algorithm is studied. Numerical results demonstrate that the HTP algorithm can effectively solve this regularization model and output very sparse solutions of LCPs with high quality.

Regularized solution of LCP problems with application to rigid body dynamics

For Linear Complementarity Problems (LCP) with a positive semidefinite matrix M, iterative solvers can be derived by a process of regularization. In [3] the initial LCP is replaced by a sequence of positive definite ones, with the matrices M + ?I. Here we analyse a generalization of this method where the identity I is replaced by a positive definite diagonal matrix D. We prove that the sequence of approximations so defined converges to the minimal D-norm solution of the initial LCP. This extension opens the possibility for interesting applications in the field of rigid multibody dynamics.

MSSOR-based alternating direction method for symmetric positive-definite linear complementarity problems

Finding the solution of linear complementarity problem (LCP) is an important subject in scientific computing, engineering and economic applications. In the literature, many iterative methods have been proposed, especially, when the matrix of the problem is a real positive definite or an H +? matrix. In this paper, we propose a new method using an alternating direction (AD) technique based on a modified symmetric successive overrelaxation (MSSOR) for LCP, assuming that the real matrix of the LCP is symmetric positive definite. This approach is an inner/outer iteration method, in which AD method is used as an outer iteration, and the MSSOR method as an inner one. We provide a detailed proof for the convergence of the proposed method. In addition, numerical results show that the proposed method is feasible and effective.

Minimal Zero Norm Solutions of Linear Complementarity Problems

In this paper, we study minimal zero norm solutions of the linear complementarity problems, defined as the solutions with smallest cardinality. Minimal zero norm solutions are often desired in some real applications such as bimatrix game and portfolio selection. We first show the uniqueness of the minimal zero norm solution for Z-matrix linear complementarity problems. To find minimal zero norm solutions is equivalent to solve a difficult zero norm minimization problem with linear complementarity constraints. We then propose a p norm regularized minimization model with p in the open interval from zero to one, and show that it can approximate minimal zero norm solutions very well by sequentially decreasing the regularization parameter. We establish a threshold lower bound for any nonzero entry in its local minimizers, that can be used to identify zero entries precisely in computed solutions. We also consider the choice of regularization parameter to get desired sparsity. Based on the theoretical results, we design a sequential smoothing gradient method to solve the model. Numerical results demonstrate that the sequential smoothing gradient method can effectively solve the regularized model and get minimal zero norm solutions of linear complementarity problems.

A shrinkage-thresholding projection method for sparsest solutions of LCPs

In this paper, we study the sparsest solutions of linear complementarity problems (LCPs), which study has many applications, such as bimatrix games and portfolio selections. Mathematically, the underlying model is NP-hard in general. By transforming the complementarity constraints into a fixed point equation with projection type, we propose an l 1 regularization projection minimization model for relaxation. Through developing a thresholding representation of solutions for a key subproblem of this regularization model, we design a shrinkage-thresholding projection (STP) algorithm to solve this model and also analyze convergence of STP algorithm. Numerical results demonstrate that the STP method can efficiently solve this regularized model and get a sparsest solution of LCP with high quality.MSC:90C33, 90C26, 90C90.

Modification of iterative methods for solving linear complementarity problems

Purpose – The purpose of this paper is to present the efficient iterative methods for solving linear complementarity problems (LCP), using a class of pre-conditioners. Design/methodology/approach – By using the concept of solving the fixed-point system of equations associated to the LCP, pre-conditioning techniques and Krylov subspace methods the authors design some projected methods to solve LCP. Furthermore, within the computational framework, some models of pre-conditioners candidates are investigated and evaluated. Findings – The proposed algorithms have a simple and graceful structure and can be applied to other complementarity problems. Asymptotic convergence of the sequence generated by the method to the unique solution of LCP is proved, along with a result regarding the convergence rate of the pre-conditioned methods. Finally, a computational comparison of the standard methods against pre-conditioned methods based on Example 1 is presented which illustrate the merits of simplicity, power and effectiveness of the proposed algorithms. Research limitations/implications – Comparison between the authors' methods and other similar methods for the studied problem shows a remarkable agreement and reveals that their models are superior in point of view rate of convergence and computing efficiency. Originality/value – For solving LCP more attention has recently been paid on a class of iterative methods called the matrix-splitting such as AOR, MAOR, GAOR, SSOR, etc. But up to now, no paper has discussed the effect of pre-conditioning technique for matrix-splitting methods in LCP. So, this paper is planning to fill in this gap and the authors use a class of pre-conditioners with iterative methods and analyze the convergence of these methods for LCP.

A full-Newton step feasible weighted primal-dual interior point algorithm for monotone LCP

In this paper, we propose a weighted short-step primal-dual interior point algorithm for solving monotone linear complementarity problem (LCP). The algorithm uses at each interior point iteration a full-Newton step and the strategy of the central path to obtain an \(\epsilon \)-approximate solution of LCP. This algorithm yields the best currently well-known theoretical complexity iteration bound, namely, \(O(\sqrt{n}\log \frac{n}{\epsilon })\) which is as good as the bound for the linear optimization analogue. The implementation of the algorithm and the algorithm in Wang et al. (Fuzzy Inform Eng 54:479–487, 2009) is done followed by a comparison between these two obtained numerical results.