Horizontal Linear Complementarity Problem Research Articles

A characteristic quantity of P-matrices

In this note, we develop some new properties of a fundamental quantity associated with a P-matrix introduced by Mathias and Pang [1]. Also, based on extensions of such a quantity, we obtain global error bounds for the vertical and horizontal linear complementarity problems.

A smoothing Gauss–Newton method for the generalized HLCP

In this paper, we present a smoothing Gauss–Newton method for solving the generalized horizontal linear complementarity problem and prove that the method is both globally and locally quadratically convergent under reasonable assumptions. As a by-product of our analysis, we obtain a sufficient condition for the existence and boundedness of the solutions to the problem.

Local Uniqueness of Solutions to the Extended Linear Complementarity Problem

In this paper, we study the local uniqueness of the solutions to the extended linear complementarity problem (XLCP, Ref. 1) by means of a concept which is an extension of the nondegenerate matrix in the standard LCP. Then, we give some special characterizations for the local uniqueness of the solutions to the horizontal linear complementarity problem (HLCP).

Some Optimization Reformulations of the Extended Linear Complementarity Problem

We consider the extended linear complementarity problem (XLCP) introduced by Mangasarian and Pang [22], of which the horizontal and vertical linear complementarity problems are two special cases. We give some new sufficient conditions for every stationary point of the natural bilinear program associated with XLCP to be a solution of XLCP. We further propose some unconstrained and bound constrained reformulations for XLCP, and study the properties of their stationary points under assumptions similar to those for the bilinear program.

High Order Infeasible-Interior-Point Methods for Solving Sufficient Linear Complementarity Problems

In this paper we develop systematically infeasible-interior-point methods of arbitrarily high order for solving horizontal linear complementarity problems that are sufficient in the sense of Cottle, Pang and Venkateswaran (1989). The results apply to degenerate problems and problems having no strictly complementary solution. Variants of these methods are described that eventually avoid recentering steps, and for which all components of the approximate solutions converge superlinearly at a high order, and other variants which even terminate with a solution of the complementarity problem after finitely many steps.

Infeasible-interior-point paths for sufficient linear complementarity problems and their analyticity

In this paper we study the behavior of infeasible-interior-point-paths for solving horizontal linear complementarity problems that are sufficient in the sense of Cottle et al. (R. W. Cottle, J.-S. Pang, Venkateswaran, Linear Algebra Appl. 114/115 (1989) 231–249). We show that these paths converge to a central point of the set of solutions. It is also shown that these are analytic functions of the path parameter even at the limitpoint, if the complementarity problem has a strictly complementary solution, and have a simple branchpoint, if it is solveable, but has no strictly complementarity solution.

An infeasible-interior-point predictor-corrector algorithm for theP *-geometric LCP

AP *-geometric linear complementarity problem (P *GP) as a generalization of the monotone geometric linear complementarity problem is introduced. In particular, it contains the monotone standard linear complementarity problem and the horizontal linear complementarity problem. Linear and quadratic programming problems can be expressed in a “natural” way (i.e., without any change of variables) asP *GP. It is shown that the algorithm of Mizunoet al. [6] can be extended to solve theP *GP. The extended algorithm is globally convergent and its computational complexity depends on the quality of the starting points. The algorithm is quadratically convergent for problems having a strictly complementary solution.

On the Convergence of the Iteration Sequence of Infeasible Path Following Algorithms for Linear Complementarity Problems

A generalized class of infeasible-interior-point methods for solving horizontal linear complementarity problems is analyzed and sufficient conditions are given for the convergence of the sequence of iterates produced by methods in this class. In particular it is shown that the largest step path following algorithms generates convergent iterates even when starting from infeasible points. The computational complexity of the latter method is discussed in detail and its local convergent rate is analyzed. The primal-dual gap of the iterates produced by this method is superlinearly convergent to zero. A variant of the method has quadratic convergence.

Equivaence between different formulations of the linear complementarity promblem

One shows that different formulations of the linear complementarity problem (LCP), such as the horizontal LCP, the mixed LCP and the geometric LCP can be transformed into a standard LCR The P*(k)-property (a more general property than monotonicity) of the corresponding formulations as well as the convergence properties of a large class of interior-point algorithms are invariant with respect to the transformations. Therefore it is sufficient to study the algorithms only for the standard LCP

Limiting Behavior of the Derivatives of Certain Trajectories Associated with a Monotone Horizontal Linear Complementarity Problem

Given a certain monotone horizontal linear complementarity problem (HLCP), we can naturally construct a family of systems of nonlinear equations parametrized by a parameter t ∈ (0, 1] with the property that, as t tends to 0, the corresponding system “converges” to the HLCP. Under reasonable conditions, it has been shown that each system of the family has a unique solution and that, as t tends to 0, these solutions converge to a specific solution of the HLCP. The main purpose of this paper is to study the asymptotic behavior of the derivative of the trajectory of solutions and therefore obtain information on the way the trajectory approaches the solution set of the HLCP. We show that the trajectory of solutions converges to the solution set along a unique and well-characterized direction. Moreover, if the HLCP has a solution satisfying strict complementarity then the direction forms a definite angle with any face of the feasible region which contains the limit point; otherwise, the direction is tangent to some face of the feasible region.

Convergence of Interior Point Algorithms for the Monotone Linear Complementarity Problem

The literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values, but very little is known about the convergence of the iterate sequences. This paper studies the horizontal linear complementarity problem, and derives general convergence properties for algorithms based on Newton iterations. This problem provides a simple and general framework for most existing primal-dual interior point methods. The conclusion is that most of the published algorithms of this kind generate convergent sequences. In many cases (whenever the convergence is not too fast in a certain sense), the sequences converge to the analytic center of the optimal face.

On the extended linear complementarity problem

For the extended linear complementarity problem (Mangasarian and Pang, 1995), we introduce and characterize column-sufficiency, row-sufficiency andP-properties. These properties are then specialized to the vertical, horizontal and mixed linear complementarity problems.

Generalizations of P 0- and P-properties; extended vertical and horizontal linear complementarity problems

Generalizing the concept of W 0-pair of Willson, we introduce the notions of column (row) W 0- and column (row) W -properties for a set of k + 1 square matrices { M 0, M 1, …, M k } (of the same dimension), where k ⩾ 1. When k = 1 and M 0 = I, these reduce to the familiar P 0- and P-properties of a square matrix. We show that these notions are related to the extended vertical and horizontal LCPs. Specifically, we show that these notions appear in certain feasible/infeasible interior point algorithms and that the column (row) W -property is characterized by the unique solvability in extended horizontal (vertical) LCPs. As a by-product of our analysis, we show that a monotone horizontal LCP is equivalent to a (standard) LCP and that for a monotone horizontal LCP, feasibility implies solvability.

Reducing horizontal linear complementarity problems

In this note, we present an algorithm to reduce a horizontal linear complementarity problem (HLCP) to a standard linear complementarity problem (LCP) whenever it is possible to do so. In fact, the algorithm is just a specialization of the matroid intersection algorithm. We also study the properties of matrix pairs that are analogous to the positive semidefiniteness and P- and P 0-properties of a square matrix, and show that a large class of matrix pairs, including P 0-pairs, define reducible HLCPs.

Solution of linear complementarity problems using minimization with simple bounds

We define a minimization problem with simple bounds associated to the horizontal linear complementarity problem (HLCP). When the HLCP is solvable, its solutions are the global minimizers of the associated problem. When the HLCP is feasible, we are able to prove a number of properties of the stationary points of the associated problem. In many cases, the stationary points are solutions of the HLCP. The theoretical results allow us to conjecture that local methods for box constrained optimization applied to the associated problem are efficient tools for solving linear complementarity problems. Numerical experiments seem to confirm this conjecture.

The Extended Linear Complementarity Problem

We consider an extension of the horizontal linear complementarity problem, which we call the extended linear complementarity problem (XLCP). With the aid of a natural bilinear program, we establish various properties of this extended complementarity problem; these include the convexity of the bilinear objective function under a monotonicity assumption, the polyhedrality of the solution set of a monotone XLCP, and an error bound result for a nondegenerate XLCP. We also present a finite, sequential linear programming algorithm for solving the nonmonotone XLCP.

On polynomiality of the Mehrotra-type predictor—corrector interior-point algorithms

Recently, Mehrotra [3] proposed a predictor—corrector primal—dual interior-point algorithm for linear programming. At each iteration, this algorithm utilizes a combination of three search directions: the predictor, the corrector and the centering directions, and requires only one matrix factorization. At present, Mehrotra's algorithmic framework is widely regarded as the most practically efficient one and has been implemented in the highly successful interior-point code OB1 [2]. In this paper, we study the theoretical convergence properties of Mehrotra's interior-point algorithmic framework. For generality, we carry out our analysis on a horizontal linear complementarity problem that includes linear and quadratic programming, as well as the standard linear complementarity problem. Under the monotonicity assumption, we establish polynomial complexity bounds for two variants of the Mehrotra-type predictor—corrector interior-point algorithms. These results are summarized in the last section in a table.

Reducing a monotone horizontal LCP to an LCP

In this note, we show how to reduce a monotone horizontal linear complementarity problem to a (standard) monotone linear complementarity problem. The main steps involved in this reduction are: finding a maximal linearly independent column set of a given matrix (or equivalently, converting a matrix to its reduced row echelon form) and inverting a nonsingular matrix.

On the Convergence of a Class of Infeasible Interior-Point Methods for the Horizontal Linear Complementarity Problem

Interior-point methods require strictly feasible points as starting points. In theory, this requirement does not seem to be particularly restrictive, but it can be costly in computation. To overcome this deficiency, most existing practical algorithms allow positive but infeasible starting points and seek feasibility and optimality simultaneously. Algorithms of this type shall be called infeasible interior-point algorithms. Despite their superior performance, existing infeasible interiorpoint algorithms still lack a satisfactory demonstration of theoretical convergence and polynomial complexity. This paper studies a popular infeasible interior-point algorithmic framework that was implemented for linear programming in the highly successful interior-point code OB1 [I. J. Lustig, R. E. Marsten, and D. F. Shanno, Linear Algebra Appl., 152 (1991), pp. 191–222]. For generality, the analysis is carried out on a horizontal linear complementarily problem that includes linear and quadratic programming, as well as the standard linear complementarily problem. Under minimal assumptions, it is demonstrated that with properly controlled steps the algorithm converges at a global Q-linear rate. Moreover, with properly chosen starting points it is established the algorithm can $\varepsilon $ obtain e-feasibility and $\varepsilon $-complementarily in at most $O( n^2 \ln ( 1/\varepsilon ) )$ iterations.