A continuation method for linear complementarity problems with P 0 matrix

In this article, we propose a new continuation method for solving the linear complementarity problem (LCP). The method solves one system of linear equations and carries out only a one-line search at each iteration. The continuation method is based on a modified smoothing function. The existence and continuity of a smooth path for solving the LCP with a P ₀ matrix are discussed. We investigate the boundedness of the iteration sequence generated by our continuation method under the assumption that the solution set of the LCP is nonempty and bounded. It is shown to converge to an LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution under suitable assumption. In addition, some numerical results are also reported in this article.     

A Non-Interior Continuation Algorithm for the P0 or P* LCP with Strong Global and Local Convergence Properties

We propose a non-interior continuation algorithm for the solution of the linear complementarity problem (LCP) with a P₀ matrix. The proposed algorithm differentiates itself from the current continuation algorithms by combining good global convergence properties with good local convergence properties under unified conditions. Specifically, it is shown that the proposed algorithm is globally convergent under an assumption which may be satisfied even if the solution set of the LCP is unbounded. Moreover, the algorithm is globally linearly and locally superlinearly convergent under a nonsingularity assumption. If the matrix in the LCP is a P_* matrix, then the above results can be strengthened to include global linear and local quadratic convergence under a strict complementary condition without the nonsingularity assumption.     

Growth Behavior of Two Classes of Merit Functions for Symmetric Cone Complementarity Problems

In the solution methods of the symmetric cone complementarity problem (SCCP), the squared norm of a complementarity function serves naturally as a merit function for the problem itself or the equivalent system of equations reformulation. In this paper, we study the growth behavior of two classes of such merit functions, which are induced by the smooth EP complementarity functions and the smooth implicit Lagrangian complementarity function, respectively. We show that, for the linear symmetric cone complementarity problem (SCLCP), both the EP merit functions and the implicit Lagrangian merit function are coercive if the underlying linear transformation has the P-property; for the general SCCP, the EP merit functions are coercive only if the underlying mapping has the uniform Jordan P-property, whereas the coerciveness of the implicit Lagrangian merit function requires an additional condition for the mapping, for example, the Lipschitz continuity or the assumption as in (45). The authors would like to thank the two anonymous referees for their helpful comments which improved the presentation of this paper greatly. The research of J.-S. Chen was partially supported by National Science Council of Taiwan.     

Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R ₀-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.     

A linearly convergent derivative-free descent method for the second-order cone complementarity problem

We consider a class of derivative-free descent methods for solving the second-order cone complementarity problem (SOCCP). The algorithm is based on the Fischer–Burmeister (FB) unconstrained minimization reformulation of the SOCCP, and utilizes a convex combination of the negative partial gradients of the FB merit function ψ_FB as the search direction. We establish the global convergence results of the algorithm under monotonicity and the uniform Jordan P-property, and show that under strong monotonicity the merit function value sequence generated converges at a linear rate to zero. Particularly, the rate of convergence is dependent on the structure of second-order cones. Numerical comparisons are also made with the limited BFGS method used by Chen and Tseng (An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104(2005), pp. 293–327), which confirm the theoretical results and the effectiveness of the algorithm.     

Strict Semimonotonicity Property of Linear Transformations on Euclidean Jordan Algebras

Motivated by the equivalence of the strict semimonotonicity property of the matrix A and the uniqueness of the solution to the linear complementarity problem LCP(A,q) for q∈R ₊ ⁿ , we study the strict semimonotonicity (SSM) property of linear transformations on Euclidean Jordan algebras. Specifically, we show that, under the copositive condition, the SSM property is equivalent to the uniqueness of the solution to LCP(L,q) for all q in the symmetric cone K. We give a characterization of the uniqueness of the solution to LCP(L,q) for a Z transformation on the Lorentz cone ℒ₊ ⁿ . We study also a matrix-induced transformation on the Lorentz space ℒ ⁿ .     

Newton iterations in implicit time-stepping scheme for differential linear complementarity systems

We propose a generalized Newton method for solving the system of nonlinear equations with linear complementarity constraints in the implicit or semi-implicit time-stepping scheme for differential linear complementarity systems (DLCS). We choose a specific solution from the solution set of the linear complementarity constraints to define a locally Lipschitz continuous right-hand-side function in the differential equation. Moreover, we present a simple formula to compute an element in the Clarke generalized Jacobian of the solution function. We show that the implicit or semi-implicit time-stepping scheme using the generalized Newton method can be applied to a class of DLCS including the nondegenerate matrix DLCS and hidden Z-matrix DLCS, and has a superlinear convergence rate. To illustrate our approach, we show that choosing the least-element solution from the solution set of the Z-matrix linear complementarity constraints can define a Lipschitz continuous right-hand-side function with a computable Lipschitz constant. The Lipschitz constant helps us to choose the step size of the time-stepping scheme and guarantee the convergence.     

A Smoothing Method for a Mathematical Program with P-Matrix Linear Complementarity Constraints

We consider a mathematical program whose constraints involve a parametric P-matrix linear complementarity problem with the design (upper level) variables as parameters. Solutions of this complementarity problem define a piecewise linear function of the parameters. We study a smoothing function of this function for solving the mathematical program. We investigate the limiting behaviour of optimal solutions, KKT points and B-stationary points of the smoothing problem. We show that a class of mathematical programs with P-matrix linear complementarity constraints can be reformulated as a piecewise convex program and solved through a sequence of continuously differentiable convex programs. Preliminary numerical results indicate that the method and convex reformulation are promising.     

Global Linear and Quadratic One-step Smoothing Newton Method for P 0-LCP

We propose a new smoothing Newton method for solving the P ₀-matrix linear complementarity problem (P ₀-LCP) based on CHKS smoothing function. Our algorithm solves only one linear system of equations and performs only one line search per iteration. It is shown to converge to a P ₀-LCP solution globally linearly and locally quadratically without the strict complementarity assumption at the solution. To the best of author's knowledge, this is the first one-step smoothing Newton method to possess both global linear and local quadratic convergence. Preliminary numerical results indicate that the proposed algorithm is promising.     

Stationary point conditions for the FB merit function associated with symmetric cones

For the symmetric cone complementarity problem, we show that each stationary point of the unconstrained minimization reformulation based on the Fischer-Burmeister merit function is a solution to the problem, provided that the gradient operators of the mappings involved in the problem satisfy column monotonicity or have the Cartesian P₀-property. These results answer the open question proposed in the article that appeared in Journal of Mathematical Analysis and Applications 355 (2009) 195-215.     

New Constrained Optimization Reformulation of Complementarity Problems

We suggest a reformulation of the complementarity problem CP(F) as a minimization problem with nonnegativity constraints. This reformulation is based on a particular unconstrained minimization reformulation of CP(F) introduced by Geiger and Kanzow as well as Facchinei and Soares. This allows us to use nonnegativity constraints for all the variables or only a subset of the variables on which the function F depends. Appropriate regularity conditions ensure that a stationary point of the new reformulation is a solution of the complementarity problem. In particular, stationary points with negative components can be avoided in contrast to the reformulation as unconstrained minimization problem. This advantage will be demonstrated for a class of complementarity problems which arise when the Karush–Kuhn–Tucker conditions of a convex inequality constrained optimization problem are considered.     

Enumeration Approach for Linear Complementarity Problems Based on a Reformulation-Linearization Technique

In this paper, we consider the linear complementarity problem (LCP) and present a global optimization algorithm based on an application of the reformulation-linearization technique (RLT). The matrix M associated with the LCP is not assumed to possess any special structure. In this approach, the LCP is formulated first as a mixed-integer 0–1 bilinear programming problem. The RLT scheme is then used to derive a new equivalent mixed-integer linear programming formulation of the LCP. An implicit enumeration scheme is developed that uses Lagrangian relaxation, strongest surrogate and strengthened cutting planes, and a heuristic, designed to exploit the strength of the resulting linearization. Computational experience on various test problems is presented.     

Sub-quadratic convergence of a smoothing Newton algorithm for the <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">0</Emphasis></Subscript>– and monotone LCP

Given , the linear complementarity problem (LCP) is to find such that (x, s) 0,s=Mx+q,x^Ts=0. By using the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, the LCP is reformulated as a system of parameterized smooth-nonsmooth equations. As a result, a smoothing Newton algorithm, which is a modified version of the Qi-Sun-Zhou algorithm [Mathematical Programming, Vol. 87, 2000, pp. 1–35], is proposed to solve the LCP with M being assumed to be a P₀-matrix (P₀–LCP). The proposed algorithm needs only to solve one system of linear equations and to do one line search at each iteration. It is proved in this paper that the proposed algorithm has the following convergence properties: (i) it is well-defined and any accumulation point of the iteration sequence is a solution of the P₀–LCP; (ii) it generates a bounded sequence if the P₀–LCP has a nonempty and bounded solution set; (iii) if an accumulation point of the iteration sequence satisfies a nonsingularity condition, which implies the P₀–LCP has a unique solution, then the whole iteration sequence converges to this accumulation point sub-quadratically with a Q-rate 2–t, where t(0,1) is a parameter; and (iv) if M is positive semidefinite and an accumulation point of the iteration sequence satisfies a strict complementarity condition, then the whole sequence converges to the accumulation point quadratically.This authors work is supported by the Hong Kong Research Grant Council and the Australian Research Council.This authors work is supported by Grant R146-000-035-101 of National University of Singapore.Mathematics Subject Classification (1991): 90C33, 65K10     

Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R₀-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.     

On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian

Mangasarian and Solodov (Ref. 1) proposed to solve nonlinear complementarity problems by seeking the unconstrained global minima of a new merit function, which they called implicit Lagrangian. A crucial point in such an approach is to determine conditions which guarantee that every unconstrained stationary point of the implicit Lagrangian is a global solution, since standard unconstrained minimization techniques are only able to locate stationary points. Some authors partially answered this question by giving sufficient conditions which guarantee this key property. In this paper, we settle the issue by giving a necessary and sufficient condition for a stationary point of the implicit Lagrangian to be a global solution and, hence, a solution of the nonlinear complementarity problem. We show that this new condition easily allows us to recover all previous results and to establish new sufficient conditions. We then consider a constrained reformulation based on the implicit Lagrangian in which nonnegative constraints on the variables are added to the original unconstrained reformulation. This is motivated by the fact that often, in applications, the function which defines the complementarity problem is defined only on the nonnegative orthant. We consider the KKT-points of this new reformulation and show that the same necessary and sufficient condition which guarantees, in the unconstrained case, that every unconstrained stationary point is a global solution, also guarantees that every KKT-point of the new problem is a global solution.     

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) 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 ≥ 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.     

The global linear and local quadratic convergence of a non-interior continuation algorithm for the LCP

^** Email: zhenghaihuang{at}yahoo.com.cn; huangzhenghai{at}hotmail.com In this paper, we propose a non-interior continuation algorithmfor solving the P₀-matrix linear complementarity problem (LCP),which is conceptually simpler than most existing non-interiorcontinuation algorithms in the sense that the proposed algorithmonly needs to solve at most one linear system of equations ateach iteration. We show that the proposed algorithm is globallyconvergent under a common assumption. In particular, we showthat the proposed algorithm is globally linearly and locallyquadratically convergent under some assumptions which are weakerthan those required in many existing non-interior continuationalgorithms. It should be pointed out that the assumptions usedin our analysis of both global linear and local quadratic convergencedo not imply the uniqueness of the solution to the LCP concerned.To the best of our knowledge, such a convergence result hasnot been reported in the literature.     

Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods

A family of complementarity problems is defined as extensions of the well-known linear complementarity problem (LCP). These are:

Superlinear／Quadratic One-step Smoothing Newton Methodf or P0-NCP

We propose a one-step smoothing Newton method for solving the non-linear complementarity problem with P0-function (P0-NCP) based on the smoothing symmetric perturbed Fisher function(for short, denoted as the SSPF-function). The proposed algorithm has to solve only one linear system of equations and performs only one line search per iteration. Without requiring any strict complementarity assumption at the P0-NCP solution, we show that the proposed algorithm converges globally and superlinearly under mild conditions. Furthermore, the algorithm has local quadratic convergence under suitable conditions. The main feature of our global convergence results is that we do not assume a priori the existence of an accumulation point. Compared to the previous literatures, our algorithm has stronger convergence results under weaker conditions.     

Basic-set algorithm for a generalized linear complementarity problem

In this paper, the authors develop a new direct method for the solution of a BLCP, that is, a linear complementarity problem (LCP) with upper bounds, when its matrix is a symmetric or an unsymmetricP-matrix. The convergence of the algorithm is established by extending Murty's principal pivoting method to an LCP which is equivalent to the BLCP. Computational experience with large-scale BLCPs shows that the basic-set method can solve efficiently large-scale BLCPs with a symmetric or an unsymmetricP-matrix.