Semidefinite Linear Complementarity Problems Research Articles

Superlinear convergence of an interior point algorithm on linear semi-definite feasibility problems

In the literature, besides the assumption of strict complementarity, superlinear convergence of implementable polynomial-time interior point algorithms using known search directions, namely, the HKM direction, its dual or the NT direction, to solve semi-definite programs (SDPs) is shown by (i) assuming that the given SDP is nondegenerate and making modifications to these algorithms [Kojima et al. Local convergence of predictor–corrector infeasible-interior-point algorithms for SDPs and SDLCPs, Math. Program. Ser. A 80 (1998), pp. 129–160], or (ii) considering special classes of SDPs, such as the class of linear semi-definite feasibility problems (LSDFPs) and requiring the initial iterate to the algorithm to satisfy certain conditions [Sim, Superlinear convergence of an infeasible predictor–corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg–Kojima–Monteiro direction, SIAM J. Optim. 21 (2011), pp. 102–126], [Sim, Interior point method on semi-definite linear complementarity problems using the Nesterov-Todd (NT) search direction: Polynomial complexity and local convergence, Comput. Optim. Appl. 74 (2019), pp. 583–621]. Otherwise, these algorithms are not easy to implement even though they are shown to have polynomial iteration complexities and superlinear convergence [Luo et al. Superlinear convergence of a symmetric primal–dual path following algorithm for semidefinite programming, SIAM J. Optim. 8 (1998), pp. 59–81]. The conditions in [Sim, Superlinear convergence of an infeasible predictor–corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg–Kojima–Monteiro direction, SIAM J. Optim. 21 (2011), pp. 102–126], [Sim, Interior point method on semi-definite linear complementarity problems using the Nesterov-Todd (NT) search direction: Polynomial complexity and local convergence, Comput. Optim. Appl. 74 (2019), pp. 583–621] that the initial iterate to the algorithm is required to satisfy to have superlinear convergence when solving LSDFPs however are not practical. In this paper, we propose a practical initial iterate to an implementable infeasible interior point algorithm that guarantees superlinear convergence when the algorithm is used to solve the homogeneous feasibility model of an LSDFP.

A NEW KERNEL FUNCTION GENERATING THE BEST COMPLEXITY ANALYSIS FOR MONOTONE SDLCP

In this article, we propose a new class of search directions based on new kernel function to solve the monotone semidefinite linear complementarity problem by primal-dual interior point algorithm. We show that this algorithm based on this function benefits from the best polynomial complexity, namely $ O(\sqrt{n}(\log n)^{2}\log\dfrac{n}{\epsilon})$. The implementation of the algorithm showed a great improvement concerning the time and the number of iterations.

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.

Convergence analysis on matrix splitting iteration algorithm for semidefinite linear complementarity problems

In this paper, we present some novel observations for the semidefinite linear complementarity problems, abbreviated as SDLCPs. Based on these new results, we establish the modulus-based matrix splitting iteration methods, which are obtained by reformulating equivalently SDLCP as an implicit fixed-point matrix equation. The convergence of the proposed modulus-based matrix splitting iteration methods has been analyzed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SDLCPs.

Interior point method on semi-definite linear complementarity problems using the Nesterov\u2013Todd (NT) search direction: polynomial complexity and local convergence

We consider in this paper an infeasible predictor–corrector primal–dual path following interior point algorithm using the Nesterov–Todd search direction to solve semi-definite linear complementarity problems. Global convergence and polynomial iteration complexity of the algorithm are established. Two sufficient conditions are also given for superlinear convergence of iterates generated by the algorithm. Preliminary numerical results are finally provided when the algorithm is used to solve semi-definite linear complementarity problems.

Complexity analysis and numerical implementation of large-update interior-point methods for SDLCP based on a new parametric barrier kernel function

In this paper, we deal with the complexity analysis and the numerical implementation of primal–dual interior-point methods for monotone semidefinite linear complementarity problems based on a new parametric kernel function. The proposed kernel function is neither a self-regular and nor the usual logarithmic barrier function. By means of the feature of the parametric kernel function, we study the complexity analysis of primal–dual IPMs and derive the currently best known iteration bound for the large-update algorithm, namely, which is as good as the linear and the semidefinite optimization analogue. Finally, we report some numerical results to show the practical performance of the proposed algorithm with different parameters.

A full Nesterov-Todd step primal-dual path-following interior-point algorithm for semidefinite linear complementarity problems

A full Nesterov-Todd step primal-dual path-following interior-point algorithm for semidefinite linear complementarity problems

An infeasible interior point method for the monotone SDLCP based on a transformation of the central path

In this paper, we generalize Roos’s infeasible interior point algorithm for linear optimization (LO) (Roos in SIAM J Optim 25(1): 102–114, 2015) to the monotone semidefinite linear complementarity problem, based on Darvay et al.’s technique for LO (Darvay et al. in Period Math Hung 73(1): 27–42, 2016). The symmetrization of the search directions is based on the Nesterov–Todd scaling scheme. The algorithm uses only one full step as feasibility step at each iteration. The derived complexity bound coincides with the best obtained one for infeasible interior-point methods with small updates.

Extensions of P-property, R0-property and semidefinite linear complementarity problems

In this manuscript, we present some new results for the semidefinite linear complementarity problem, in the context of three notions for linear transformations, viz., pseudo w-P property, pseudo Jordan w-P property and pseudo SSM property. Interconnections with the P#-property (proposed recently in the literature) is presented. We also study the R#-property of a linear transformation, extending the rather well known notion of an R0-matrix. In particular, results are presented for the Lyapunov, Stein, and the multiplicative transformations

An active set algorithm for a class of linear complementarity problems arising from rigid body dynamics

An active set algorithm is introduced for positive definite and positive semi definite linear complementarity problems. The proposed algorithm is composed of two phases. Phase 1, the feasibility phase and phase 2, the optimality phase. In phase 1, the ellipsoid method is employed to test for feasibility and provide an advanced starting point if the problem is feasible. Providing such a warm start permits a good estimate of the active set. In phase 2, a criterion based on the complementarity condition is used to detect the working set per iteration until optimality is reached. This criterion leads to a valuable reduction in the size of the problem solved per iteration to obtain a search direction. Numerical examples are solved to illustrate the performance of the algorithm and a practical example in rigid body dynamics is solved to demonstrate the usage of the algorithm to solve such problems.

Low-order penalty equations for semidefinite linear complementarity problems

We extend the power penalty method for linear complementarity problem (LCP) (Wang and Yang, 2008) to the semidefinite linear complementarity problem (SDLCP). We establish a family of low-order penalty equations for SDLCPs. Under the assumption that the involved linear transformation possesses the Cartesian P-property, we show that when the penalty parameter tends to infinity, the solution to any equation of this family converges to the solution of the SDLCP exponentially. Numerical experiments verify this convergence result.

A modified homogeneous potential reduction algorithm for solving the monotone semidefinite linear complementarity problem

In this paper, we study the performance of the Semidefinite Linear Complementarity Problem (SDLCP) for symmetric matrices that is equipped with a continuously differentiable potential function. A practical homogeneous self-dual potential reduction algorithm based on this potential function is prescribed, and we establish a computational basis for interior point methods with the use of HKM directions towards the central trajectory for the monotone SDLCP. Our computational implementation maintains a global linear polynomial time convergence, while achieving strong practical performance in comparison with existing solution methods.

SOLVING STRONGLY MONOTONE LINEAR COMPLEMENTARITY PROBLEMS

Given a linear transformation L on a finite dimensional real inner product space V to itself and an element q ∈ V we consider the general linear complementarity problem LCP (L, K, q) on a proper cone K ⊆ V. We observe that the iterates generated by any closed algorithmic map will converge to a solution for LCP (L, K, q), whenever L is strongly monotone. Lipschitz constants of L is vital in establishing the above said convergence. Hence we compute the Lipschitz constants for certain classes of Lyapunov, Stein and double-sided multiplicative transformations in the setting of semidefinite linear complementarity problems. We give a numerical illustration of a closed algorithmic map in the setting of a standard linear complementarity problem. On account of the difficulties in numerically implementing such algorithms for general linear complementarity problems, we give an alternative algorithm for computing the solution for a special class of strongly monotone semidefinite linear complementarity problems along with a numerical example.

A class of singular Ro-matrices and extensions to semidefinite linear complementarity problems

For ARnxn and qRn, the linear complementarity problem LCP(A, q) is to determine if there is xRn such that x ≥ 0; y = Ax + q ≥ 0 and xT y = 0. Such an x is called a solution of LCP(A,q). A is called an Ro-matrix if LCP(A,0) has zero as the only solution. In this article, the class of R0-matrices is extended to include typically singular matrices, by requiring in addition that the solution x above belongs to a subspace of Rn. This idea is then extended to semidefinite linear complementarity problems, where a characterization is presented for the multplicative transformation.

Linear Complementarity Problems over Symmetric Cones: Characterization of Q b -transformations and Existence Results

This paper is devoted to the study of the symmetric cone linear complementarity problem (SCLCP). Specifically, our aim is to characterize the class of linear transformations for which the SCLCP has always a nonempty and bounded solution set in terms of larger classes. For this, we introduce a couple of new classes of linear transformations in this SCLCP context. Then, we study them for concrete particular instances (such as second-order and semidefinite linear complementarity problems) and for specific examples (Lyapunov, Stein functions, among others). This naturally permits to establish coercive and noncoercive existence results for SCLCPs.

Plynomial Convergence of Primal-Dual Algorithms for SDLCP Based on the Monteiro-Zhang Family of Directions

The paper establishes the polynomial converge-nce of a new class of path-following methods for semide- finite linear complementarity problems (SDLCP) whos-se search directions belong to the class of directions introduced by Monteiro [7]. Namely, we show that the polynomial iteration complexity bounds of the well known algori-thm for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Alder, carry over to the context of SDLCP

Minimum norm solution to the positive semidefinite linear complementarity problem

In this article, we present an algorithm to compute the minimum norm solution of the positive semidefinite linear complementarity problem. We show that its solution can be obtained using the alternative theorems and a convenient characterization of the solution set of a convex quadratic programming problem. This problem reduces to an unconstrained minimization problem with once differentiable convex objective function. We propose an extension of Newton's method for solving the unconstrained optimization problem. Computational results show that convergence to high accuracy often occurs in just a few iterations.

Complexity analysis of primal-dual algorithms for the semidefinite linear complementarity problem

In this paper a primal-dual path-following interior-point algorithm for the monotone semidefinite linear complementarity problem is presented. The algorithm is based on Nesterov-Todd search directions and on a suitable proximity for tracing approximately the central-path. We provide an unified analysis for both long and small-update primal-dual algorithms. Finally, the iteration bounds for these algorithms are obtained.

Superlinear Convergence of an Infeasible Predictor-Corrector Path-Following Interior Point Algorithm for a Semidefinite Linear Complementarity Problem Using the Helmberg–Kojima–Monteiro Direction

An interior point method (IPM) defines a search direction at each interior point of a region. These search directions form a direction field which in turn gives rise to a system of ordinary differential equations (ODEs). The solutions of the system of ODEs can be viewed as underlying paths in the interior of the region. In [C.-K. Sim and G. Zhao, Math. Program. Ser. A, 110 (2007), pp. 475-499], these off-central paths are shown to be well-defined analytic curves, and any of their accumulation points is a solution to a given monotone semidefinite linear complementarity problem (SDLCP). The study of these paths provides a way to understand how iterates generated by an interior point algorithm behave. In this paper, we give a sufficient condition using these off-central paths that guarantees superlinear convergence of a predictor-corrector path-following interior point algorithm for SDLCP using the Helmberg-Kojima-Monteiro (HKM) direction. This sufficient condition is implied by a currently known sufficient condition for superlinear convergence. Using this sufficient condition, we show that for any linear semidefinite feasibility problem, superlinear convergence using the interior point algorithm, with the HKM direction, can be achieved for a suitable starting point. We work under the assumption of strict complementarity.

Asymptotic Behavior of Underlying NT Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems

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 solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), 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). In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007; J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of off-central paths corresponding to the HKM direction is studied. In particular, in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), the authors study the asymptotic behavior of these paths for a simple example, while, in Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of these paths for a general SDLCP is studied. In this paper, we study off-central paths corresponding to another well-known direction, the Nesterov-Todd (NT) direction. Again, we give necessary and sufficient conditions for these off-central paths to be analytic w.r.t. \(\sqrt{\mu}\) and then w.r.t. μ, at solutions of a general SDLCP. Also, as in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), we present off-central path examples using the same SDP, whose first derivatives are likely to be unbounded as they approach the solution of the SDP. We work under the assumption that the given SDLCP satisfies a strict complementarity condition.