Semidefinite Linear Complementarity Problems Research Articles

Characterization of [formula omitted]-property for multiplicative transformations in semidefinite linear complementarity problems

We characterize the Q -property of a multiplicative transformation in semidefinite linear complementarity problems.

Shifted skew-symmetric iteration methods for nonsymmetric linear complementarity problems

We present a shifted skew-symmetric iteration method for solving the nonsymmetric positive definite or positive semidefinite linear complementarity problems. This method is based on the symmetric and skew-symmetric splitting of the system matrix, which has been adopted to establish efficient splitting iteration methods for solving the nonsymmetric systems of linear equations. Global convergence of the method is proved, and the corresponding inexact splitting iteration scheme is established and analyzed in detail. Numerical results show that the new methods are feasible and effective for solving large sparse and nonsymmetric linear complementarity problems.

On the [formula omitted] and [formula omitted]-properties in the semidefinite linear complementarity problem

Motivated by the so-called P2-property in the semidefinite linear complementarity problems, in this article, we introduce the concept of P2′-property for a linear transformation on the space of real n×n symmetric matrices. While these two properties turn out to be different, we show that they are equivalent for the Lyapunov transformation LA, double-sided multiplicative transformation MA and a particular class of Stein transformations. We also show that P2′ implies the SSM and Q-properties.

On the Analyticity of Underlying HKM Paths for Monotone Semidefinite Linear Complementarity Problems

An interior point method (IPM) defines a search direction at an interior point of the feasible region. These search directions form a direction field, which in turn defines a system of ordinary differential equations (ODEs). The solutions of the system of ODEs are called off-central paths, underlying paths lying in the interior of the feasible region. It is known that not all off-central paths are analytic, whether w.r.t. μ or \(\sqrt{\mu}\) , where μ represents the duality gap, at a solution of a given semidefinite linear complementarity problem, SDLCP (Sim and Zhao, Math. Program. 110:475–499, 2007). In Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008), we give a necessary and sufficient condition for when an off-central path is analytic as a function of \(\sqrt{\mu}\) at a solution of a general SDLCP. It is then natural to ask about the analyticity of a SDLCP off-central path at a solution, as a function of μ. We investigate this in the current paper. Again, we work under the assumption that the given SDLCP satisfies strict complementarity condition.

Asymptotic Behavior of Helmberg-Kojima-Monteiro (HKM) Paths in Interior-Point Methods for Monotone Semidefinite Linear Complementarity Problems: General Theory

An interior-point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as the solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A, [2007], to appear), these off-central paths are shown to be well-defined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). Off-central paths for a simple example are also studied in Sim and Zhao (Math. Program. Ser. A, [2007], to appear) and their asymptotic behavior near the solution of the example is analyzed. In this paper, which is an extension of Sim and Zhao (Math. Program. Ser. A, [2007], to appear), we study the asymptotic behavior of the off-central paths for general SDLCPs using the dual HKM direction. We give a necessary and sufficient condition for when an off-central path is analytic as a function of \(\sqrt{\mu}\) at a solution of the SDLCP. Then, we show that, if the given SDLCP has a unique solution, the first derivative of its off-central path, as a function of \(\sqrt{\mu}\) , is bounded. We work under the assumption that the given SDLCP satisfies the strict complementarity condition.

Limiting behavior of the Alizadeh–Haeberly–Overton weighted paths in semidefinite programming

This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming obtained from centrality equations of the form X S+SX=2 ν W, where W is a fixed positive definite matrix and ν>0 is a parameter, under the assumption that the problem has a strictly complementary primal–dual optimal solution. We present a different and simpler proof than the one given by Preiß and Stoer [Preiß, M. and Stoer, J., 2004, Analysis of infeasible-interior-point paths arising with semidefinite linear complementarity problems. Mathematical Programming 99, 499–520.] that a weighted central path as a function of ν can be extended analytically beyond 0. In addition, the characterization of the limit points of the path and its normalized first-order derivatives is also provided. We also derive an error bound on the distance between a point lying in a certain neighborhood of the central path and the set of primal–dual optimal solutions. Finally, we make some observation for the superlinear convergence of some primal–dual interior point SDP algorithms using AHO neighborhood.

An exterior point algorithm for some linear complementarity problems with applications

An exterior point algorithm for positive definite (PD) and positive semidefinite (PSD) linear complementarity problems (LCPs) is introduced. The algorithm exploits the ellipsoid method to find a starting point in the case of positive definite linear complementarity problems (PDLCPs) and to check for the problem feasibility in case of positive semidefinite linear complementarity problems (PSDLCPs). The algorithm starts from a point on the boundary on which the complementarity condition is satisfied and generates a sequence of points on that same boundary. These points converge to the solution. The algorithm is modified to speed up the convergence for some PDLCPs and PSDLCPs that arise in certain mechanical models. A numerical example and a practical example in robotics are solved to test the algorithm.

The Q-property of a multiplicative transformation in semidefinite linear complementarity problems

The Q-property of a multiplicative transformation AXAT in semidefinite linear complementarity problems is characterized when A is normal.

Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

An interior point method defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves off-central paths. We study off-central paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each off-central path is a well-defined analytic curve with parameter μ ranging over (0, ∞) and any accumulation point of the off-central path is a solution to SDLCP. Through a simple example we show that the off-central paths are not analytic as a function of $$\sqrt{\mu}$$ and have first derivatives which are unbounded as a function of μ at μ = 0 in general. On the other hand, for the same example, we can find a subset of off-central paths which are analytic at μ = 0. These “nice” paths are characterized by some algebraic equations.

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.

Cartesian P-property and Its Applications to the Semidefinite Linear Complementarity Problem

We introduce a Cartesian P-property for linear transformations between the space of symmetric matrices and present its applications to the semidefinite linear complementarity problem (SDLCP). With this Cartesian P-property, we show that the SDLCP has GUS-property (i.e., globally unique solvability), and the solution map of the SDLCP is locally Lipschitzian with respect to input data. Our Cartesian P-property strengthens the corresponding P-properties of Gowda and Song [15] and allows us to extend several numerical approaches for monotone SDLCPs to solve more general SDLCPs, namely SDLCPs with the Cartesian P-property. In particular, we address important theoretical issues encountered in those numerical approaches, such as issues related to the stationary points in the merit function approach, and the existence of Newton directions and boundedness of iterates in the non-interior continuation method of Chen and Tseng [6].

On the Lipschitz Continuity of the Solution Map in Semidefinite Linear Complementarity Problems

In this paper, we investigate the Lipschitz continuity of the solution map in semidefinite linear complementarity problems. For a monotone linear transformation defined on the space of real symmetric n × n matrices, we show that the Lipschitz continuity of the solution map implies the globally uniquely solvable (GUS)-property. For Lyapunov transformations with the Q-property, we prove that the Lipschitz continuity of the solution map is equivalent to the strong monotonicity property. For the double-sided multiplicative transformations, we show that the Lipschitz continuity of the solution map implies the GUS-property.

A Note on the Local Convergence of a Predictor-Corrector Interior-Point Algorithm for the Semidefinite Linear Complementarity Problem Based on the Alizadeh--Haeberly--Overton Search Direction

This note points out an error in the local quadratic convergence proof of the predictor-corrector interior-point algorithm for solving the semidefinite linear complementarity problem based on the Alizadeh--Haeberly--Overton search direction presented in [M. Kojima, M. Shida, and S. Shindoh, SIAM J. Optim., 9 (1999), pp. 444--465]. Their algorithm is slightly modified and the local quadratic convergence of the resulting method is established.

Analysis of infeasible-interior-point paths arising with semidefinite linear complementarity problems

We consider semidefinite monotone linear complementarity problems (SDLCP) in the space *** equation here *** n of real symmetric n×n-matrices equipped with the cone *** equation here *** n+ of all symmetric positive semidefinite matrices. One may define weighted (using any M∈ *** equation here *** n++ as weight) infeasible interior point paths by replacing the standard condition XY=rI, r>0, (that defines the usual central path) by (XY+YX)/2=rM. Under some mild assumptions (the most stringent is the existence of some strictly complementary solution of (SDLCP)), these paths have a limit as r↓0, and they depend analytically on all path parameters (such as r and M), even at the limit point r=0.

On some interconnections between strict monotonicity, globally uniquely solvable, and P properties in semidefinite linear complementarity problems

In the setting of semidefinite linear complementarity problems on S n , the implications strict monotonicity⇒ P 2⇒ GUS ⇒ P are known. Here, P and P 2 properties for a linear transformation L: S n→ S n are respectively defined by: X∈ S n, XL( X)= L( X) X⪯0⇒ X=0 and X⪰0, Y⪰0, ( X− Y)[ L( X)− L( Y)]( X+ Y)⪯0⇒ X= Y; GUS refers to the global unique solvability in semidefinite linear complementarity problems correspond- ing to L. In this article, we show that the reverse implications hold for any self-adjoint linear transformation, and for normal Lyapunov and Stein transformations. By introducing the concept of a principal subtransformation of a linear transformation, we show that L: S n→ S n has the P 2 -property if and only if for every n× n real invertible matrix Q, every principal subtransformation of L has the P -property where L(X):=Q T L(QXQ T )Q. Based on this, we show that P 2, GUS, and P properties coincide for the two-sided multiplication transfor- mation.

Relationship Between Strong Monotonicity Property, P2-Property, and the Gus-Property in Semidefinite Linear Complementarity Problems

In a recent paper on semidefinite linear complementarity problems, Gowda and Song (2000) introduced and studied the P-property, P2-property, GUS-property, and strong monotonicity property for linear transformation L: Sn → Sn, where Sn is the space of all symmetric and real n × n matrices. In an attempt to characterize the P2-property, they raised the following two questions: (i) Does the strong monotonicity imply the P2-property? (ii) Does the GUS-property imply the P2-property? In this paper, we show that the strong monotonicity property implies the P2-property for any linear transformation and describe an equivalence between these two properties for Lyapunov and other transformations. We show by means of an example that the GUS-property need not imply the P2-property, even for Lyapunov transformations.

Some New Results for the Semidefinite Linear Complementarity Problem

In this paper, we present some new results for the semidefinite linear complementarity problem (SDLCP). In the first part, we introduce the concepts of (i) nondegeneracy for a linear transformation $L:{\cal S}^n \rightarrow {\cal S}^n$ and (ii) the locally-star-like property of a solution point of an SDLCP(L,Q) for $Q\in {\cal S}^n$, and we relate them to the finiteness of the solution set of SDLCP(L,Q) as Q varies in ${\cal S}^n$. In the second part, we show that for positive stable matrices A1,. . ., Ak, the linear transformation L:=LA1 \circ L_{A_2} \circ \cdots \circ L_{A_k} $ has the Q-property where LAi(X):= AiX + XAiT. A similar result is proved for the transformation $S:=S_{A_1} \circ S_{A_2} \circ \cdots \circ S_{A_k}$, where each Ai is Schur stable and $S_{A_i}(X):=X-A_iXA_i^{T}$. We relate these results to the simultaneous stability of a finite set of matrices.

On semidefinite linear complementarity problems

Given a linear transformation L:? n →? n and a matrix Q∈? n , where ? n is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,? n +,Q) over the cone ? n + of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix A∈R n×n , we consider the linear transformation L A :? n →? n defined by L A (X):=AX+XA T and show that the P- and Q-properties for L A are equivalent to A being positive stable, i.e., real parts of eigenvalues of A are positive. As a special case of this equivalence, we deduce a theorem of Lyapunov.

Search directions in the SDP and the monotone SDLCP: generalization and inexact computation

We propose a family of directions that generalizes many directions proposed so far in feasible-interior-point methods for the SDP (semidefinite programming) and for the monotone SDLCP (semidefinite linear complementarity problem). We derive the family from the Helmberg-Rendl-Vanderbei-Wolkowicz/ Kojima-Shindoh-Hara/Monteiro direction by relaxing its centrality equation into a centrality inequality.' Using this family, we present a predictor-corrector infeasible-interior-point method to provide a theoretical basis for inexact computation of directions in primal-dual interior-point methods for the SDP.

Polynomiality of primal-dual algorithms for semidefinite linear complementarity problems based on the Kojima-Shindoh-Hara family of directions

Kojima, Shindoh and Hara proposed a family of search directions for the semidefinite linear complementarity problem (SDLCP) and established polynomial convergence of a feasible short-step path-following algorithm based on a particular direction of their family. The question of whether polynomiality could be established for any direction of their family thus remained an open problem. This paper answers this question in the affirmative by establishing the polynomiality of primal-dual interior-point algorithms for SDLCP based on any direction of the Kojima, Shindoh and Hara family of search directions. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the context of SDLCP.