Algorithm For Linear Complementarity Problem Research Articles

Sufficient Matrices: Properties, Generating and Testing

This paper investigates various aspects of sufficient matrices, one of the most relevant matrix classes introduced in connection with linear complementarity problems. We summarize the most important theoretical results and properties related to sufficient matrices. Based on these, we propose different construction rules that can be used to generate new matrices that belong to this class. A nonnegative number can be assigned to each sufficient matrix, which is called its handicap and works as a measure of sufficiency. The handicap plays a crucial role in proving convergence and complexity results for interior point algorithms for linear complementarity problems. For a particular sufficient matrix, called Csizmadia’s matrix, we give the exact value of the handicap, which is exponential in the size of the matrix. Another important topic that we address is deciding whether a matrix is sufficient. Tseng proved in 2000 that this decision problem is co-NP hard. We investigate three different algorithms for determining the sufficiency of a given matrix: Väliaho’s algorithm, a linear programming-based algorithm, and an algorithm that facilitates nonlinear programming reformulations of the definition of sufficiency. We tested the efficiency of these methods on our recently launched benchmark data set that consists of four different sets of matrices. In this paper, we give the description and most important properties of the benchmark set, which can be used in the future to compare the performance of different interior point algorithms for 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.

Interior-point algorithm for sufficient LCPs based on the technique of algebraically equivalent transformation

We present a short-step interior-point algorithm (IPA) for sufficient linear complementarity problems (LCPs) based on a new search direction. An algebraic equivalent transformation (AET) is used on the centrality equation of the central path system and Newton’s method is applied on this modified system. This technique was offered by Zsolt Darvay for linear optimization in 2002. Darvay first used the square root function as AET and in 2012 Darvay et al. applied this technique with an other transformation formed by the difference of the identity map and the square root function. We apply the AET technique with the new function to transform the central path equation of the sufficient LCPs. This technique leads to new search directions for the problem. We introduce an IPA with full Newton steps and prove that the iteration bound of the algorithm coincides with the best known one for sufficient LCPs. We present some numerical results to illustrate performance of the proposed IPA on two significantly different sets of test problems and compare it, with related, quite similar variants of IPAs.

Implementation of an Interior-point Algorithm for Linear Complementarity Problem Working in a Wide Neighborhood

Abstract In this article, we study the interior-point algorithm for solving linear complementarity problems, published by Xiaouje Ma, Hongwei Liu, Jianke Zhang and Weijie Cong from the implementation point of view. The algorithm was implemented in C++ programming language, thus supporting the effectiveness of the method. Despite the fact that the theoretical results refer only to the monotone linear complementarity problem, practical testing showed that the algorithm also works well in more general cases.

An arc-search predictor-corrector infeasible-interior-point algorithm for P∗(κ)-SCLCPs

In this paper, we propose a new arc-search predictor-corrector infeasible-interior-point algorithm for linear complementarity problems over symmetric cones with the Cartesian P∗(κ)-property (P∗(κ)-SCLCP). The proposed algorithm is based on a wide neighborhood of the central path and searches the optimizers along the ellipses that approximate the entire the central path. The algorithm uses a commutative class of search directions, which includes the Nesterov-Todd direction and the xs and sx directions. To the best of our knowledge, this is using a new strategy in the complexity analysis; we improve the theoretical complexity bound of an arc-search infeasible-interior-point method for P∗(κ)-SCLCP. Some preliminary numerical results are provided to demonstrate the computational performance of the proposed algorithm.

A path following interior-point method for linear complementarity problems over circular cones

Circular cones are a new class of regular cones that include the well-known second-order cones as a special case. In this paper, we study the algebraic structure of the circular cone and show that based on the standard inner product this cone is nonsymmetric while using the new-defined circular inner product this cone not only is symmetric but also the algebra associated with it, is a Euclidean Jordan algebra. Then, using the machinery of Euclidean Jordan algebras and the Nestrov–Todd search directions, we propose a primal-dual path-following interior-point algorithm for linear complementarity problems over the Cartesian product of the circular cones. The convergence analysis of the algorithm is shown and it is proved that this class of mathematical problems is polynomial-time solvable.

A Wide-Neighborhood Predictor-Corrector Interior-Point Algorithm for Linear Complementarity Problems

In this paper, a wide-neighborhood predictor-corrector feasible interior-point algorithm for linear complementarity problems is proposed. The algorithm is based on using the classical affine scaling direction as a part in a corrector step, not in a predictor step. The convergence analysis of the algorithm is shown, and it is proved that the algorithm has the polynomial complexity $$O\left(\sqrt{n}\log \varepsilon ^{-1}\right)$$ which coincides with the best known iteration bound for this class of mathematical problems. The numerical results indicate the efficiency of the algorithm.

The Convergence Analysis of Parallel Alternating Two-stage Iterative Algorithm for Linear Complementarity Problem

In this paper, the authors first present parallel alternating two-stage iterative Algorithm some new relaxation algorithms for solving the linear complementarity problem. And then, when the coefficient matrices are monotone or H-matrices, they establish the global convergence theory of the algorithm. The algorithm has less computational complexity and quicker velocity and is especially suitable for parallel computation of large-scale problem.

An $$\ell _{2}$$ ℓ 2 -neighborhood infeasible interior-point algorithm for linear complementarity problems

In this paper, we propose an infeasible interior-point algorithm for linear complementarity problems. In every iteration, the algorithm constructs an ellipse and searches an \(\varepsilon \)-approximate solution of the problem along the ellipsoidal approximation of the central path. The theoretical iteration-complexity of the algorithm is derived and the algorithm is proved to be polynomial with the complexity bound \(O\left(n\log \varepsilon ^{-1}\right)\) which coincides with the best known iteration bound for infeasible interior-point methods.

A Predictor–Corrector Algorithm for Monotone Linear Complementarity Problems in a Wide Neighborhood

We propose a new primal-dual interior-point predictor–corrector algorithm in Ai and Zhang’s wide neighborhood for solving monotone linear complementarity problems (LCP). Based on the understanding of this neighborhood, we use two new directions in the predictor step and in the corrector step, respectively. Especially, the use of new corrector direction also reduces the duality gap in the corrector step, which has good effects on the algorithm’s convergence. We prove that the new algorithm has a polynomial complexity of [Formula: see text], which is the best complexity result so far. In the paper, we also prove a key result for searching for the best step size along some direction. Considering local convergence, we revise the algorithm to be a variant, which enjoys both complexity of [Formula: see text] and Q-quadratical convergence. Finally, numerical result shows the effectiveness and superiority of the two new algorithms for monotone LCPs.

An infeasible interior-point method with improved centering steps for monotone linear complementarity problems

In this paper, we improve the infeasible full-Newton interior-point algorithm presented by Mansouri et al. [A full-Newton step [Formula: see text] infeasible interior-point algorithm for linear complementarity problems, Nonlinear Anal. Real World Appl. 12 (2011) 545–561] for monotone linear complementarity problems (MLCPs). In each iteration of Mansouri’s algorithm two types of full-Newton steps are used, one feasibility step and some ordinary (centering) steps. In this paper, we use a new search direction, and reduce the number of the centering steps, so that only one centering step is needed. We prove that the complexity of the algorithm is as good as the best-known complexity for infeasible interior-point methods for MLCPs.

A weighted-path-following method for symmetric cone linear complementarity problems

In this paper a weighted-path-following interior-point algorithm for linear complementarity problem over symmetric cones is proposed that uses new search directions. The complexity results of the new algorithm derived and proved that the proposed algorithm has quadratically convergent with polynomial-time. We conclude that following the central path yields to the best iteration bound in this case as well.

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.

A two-level domain decomposition algorithm for linear complementarity problem

In this paper, a two-level domain decomposition algorithm for linear complementarity problem (LCP) is introduced. Inner and outer approximation sequences to the solution of LCP are generated by the proposed algorithm. The algorithm is proved to be convergent and can reach the solution of the problem within finite steps. Some simple numerical results are presented to show the effectiveness of the proposed algorithm.

A new class of infeasible interior-point algorithm for linear complementarity problem

This paper proposes an infeasible interior-point algorithm for linear complementarity problem with full-Newton steps. The main iteration consists of a feasibility step and several centrality steps. No more than O(nlog(n/ɛ)) iterations are required for getting an ɛ-solution of the problem at hand, which coincides with the best-known bound for infeasible interior-point algorithms.

A new interior-point algorithm based on modified Nesterov–Todd direction for symmetric cone linear complementarity problem

We present a new primal-dual path-following interior-point algorithm for linear complementarity problem over symmetric cones. The algorithm is based on a reformulation of the central path for finding the search directions. For a full Nesterov–Todd step feasible interior-point algorithm based on the search directions, the complexity bound of the algorithm with the small update approach is the best available.

A full-step interior-point algorithm for linear complementarity problem based on a simple function

In this paper, we design and analyze a new full-step interior-point algorithm for linear complementarity problem based on a very simple function. The algorithm uses the simple function to determine the searching direction and define the neighborhood of central path. The full-step used in the algorithm has local quadratic convergence property according to the proximity function which is also constructed by this simple function. We derive the iterationcomplexity for the algorithm and obtain the best-known iteration bounds for linear complementarity problem

A New Searching Direction for Linear Complementarity Problems

In this paper, we investigate the properties of a simple function. As an application, we present a full-step interior-point algorithm for linear complementarity problem. The algorithm uses the simple function to determine the searching direction and define the neighborhood of central path. The full-step used in the algorithm has local quadratic convergence property according to the proximity function which is also constructed by this simple function. We derive the iteration complexity for the algorithm and obtain the best-known iteration bounds for linear complementarity problem.

Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix

The plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) $${0\leq x\perp(Mx+q)\geq0}$$ can be viewed as a semismooth Newton algorithm without globalization technique to solve the system of piecewise linear equations min(x, Mx + q) = 0, which is equivalent to the LCP. When M is an M-matrix of order n, the algorithm is known to converge in at most n iterations. We show in this paper that this result no longer holds when M is a P-matrix of order ź 3, since then the algorithm may cycle. P-matrices are interesting since they are those ensuring the existence and uniqueness of the solution to the LCP for an arbitrary q. Incidentally, convergence occurs for a P-matrix of order 1 or 2.

Generalized AOR Algorithms for Linear Complementarity Problem

In this paper, we propose a class of generalized accelerated overrelaxation (GAOR) methods, in which one special case reduces to the generalized successive overrelaxation(GSOR) methods, for solving a linear complementarity problem LCP( M, q). Sufficient conditions for convergence of the GAOR and GSOR methods are presented for cases when the system matrix M is an H–matrix, M–matrix and a strictly or irreducible diagonally dominant matrix. Moreover, when M is an L–matrix, we examine the monotone convergence of the new methods. Finally a numerical example is used to validate the results proved in this paper.