Second-order Cone Linear Complementarity Problem Research Articles

Solving symmetric and positive definite second-order cone linear complementarity problem by a rational Krylov subspace method

The second-order cone linear complementarity problem (SOCLCP) is a generalization of the classical linear complementarity problem. It has been known that SOCLCP, with the globally uniquely solvable property, is essentially equivalent to a zero-finding problem in which the associated function bears much similarity to the transfer function in model reduction [38]. In this paper, we propose a new rational Krylov subspace method to solve the zero-finding problem for the symmetric and positive definite SOCLCP. The algorithm consists of two stages: first, it relies on an extended Krylov subspace to obtain an approximation of the zero root, and then applies multiple-pole rational Krylov subspace projections iteratively to acquire an accurate solution. Numerical evaluations on various types of SOCLCP examples demonstrate its efficiency and robustness.

PRIMAL-DUAL ALGORITHM FOR QUASI-STATIC CONTACT PROBLEM WITH COULOMB'S FRICTION

This paper presents a fast first-order method for solving the quasi-static contact problem with the Coulomb friction. It is known that this problem can be formulated as a second-order cone linear complementarity problem, for which regularized or semi-smooth Newton methods are widely used. As an alternative approach, this paper develops a method based on an accelerated primal-dual algorithm. The proposed method is easy to implement, as most of computation consists of additions and multiplications of vectors and matrices. Numerical experiments demonstrate that this method outperforms a regularized and smoothed Newton method for the second-order cone complementarity problem.

An approximate lower order penalty approach for solving second-order cone linear complementarity problems

An approximate lower order penalty approach for solving second-order cone linear complementarity problems

Characterization of Q-property for cone automorphisms in second-order cone linear complementarity problems

Let be the second-order cone in , where . Given a vector and an matrix G, the second order cone linear complementarity problem is to find a vector such that The matrix G is said to have the -property if has a solution for all . An matrix G is called a cone automorphism if . In this paper, we obtain a simple characterization for the -property of a cone automorphism. This says that G has the -property if and only if zero is the only solution to .

On matrix characterizations for P-property of the linear transformation in second-order cone linear complementarity problems

The P-property of the linear transformation in second-order cone linear complementarity problems (SOCLCP) plays an important role in checking the globally uniquely solvable (GUS) property due to the work of Gowda et al. However, it is not easy to verify the P-property of the linear transformation, in general. In this paper, we provide matrix characterizations for checking the P-property, which is a new approach different from those in the literature. This is a do-able manipulation, which helps verifications of the P-property and globally uniquely solvable (GUS) property in second-order cone linear complementarity problems. Moreover, using an equivalence relation to the second-order cone linear complementarity problem, we study some sufficient and necessary conditions for the unique solution of the absolute value equations associated with second-order cone (SOCAVE).

SOR-Like Iteration Methods for Second-Order Cone Linear Complementarity Problems

SOR-Like Iteration Methods for Second-Order Cone Linear Complementarity Problems

An Efficient Numerical Method for the Symmetric Positive Definite Second-Order Cone Linear Complementarity Problem

An efficient numerical method for solving a symmetric positive definite second-order cone linear complementarity problem (SOCLCP) is proposed. The method is shown to be more efficient than recently developed iterative methods for small-to-medium sized and dense SOCLCP. Therefore it can serve as an excellent core computational engine in solutions of large scale symmetric positive definite SOCLCP solved by subspace projection methods, solutions of general SOCLCP and the quadratic programming over a Cartesian product of multiple second-order cones, in which small-to-medium sized SOCLCPs have to be solved repeatedly, efficiently, and robustly.

The modulus-based matrix splitting iteration methods for second-order cone linear complementarity problems

For the second-order cone linear complementarity problems, abbreviated as SOCLCPs, we establish two classes of modulus-based matrix splitting iteration methods, which are obtained by reformulating equivalently the SOCLCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The convergence of these modulus-based matrix splitting iteration methods has been established and the optimal iteration parameters of these methods are discussed when the splitting matrix is symmetric positive definite. Numerical experiments have shown that the modulus-based iteration methods are effective for solving the SOCLCPs.

Conditions for Error Bounds of Linear Complementarity Problems over Second-Order Cones with Pseudomonotonicity

We first discuss some properties of the solution set of a pseudomonotone second-order cone linear complementarity problem (SOCLCP), and then analyse the limiting behavior of a sequence of strictly feasible solutions within a new wide neighborhood of the central trajectory for the pseudomonotone SOCLCP under assumptions of strict complementarity. Based on this, we derive four different characterizations of an error bound for the pseudomonotone SOCLCP.

On the Range of the Pseudomonotone Second-Order Cone Linear Complementarity Problem

In this paper, we provide a complete characterization of the range of the pseudomonotone second-order cone linear complementarity problem. In particular, by answering the three questions that under what conditions the range is the whole space, convex and closed, respectively, we explicitly characterize and formulate the range of the pseudomonotone second-order cone linear complementarity problem.

On the Lipschitz continuity of the solution map in linear complementarity problems over second-order cone

Let K⊆IRn denote the second-order cone. Given an n×n real matrix M and a vector q∈IRn, the second-order cone linear complementarity problem SOLCP(M,q) is to find a vector x∈IRn such thatx∈K,y:=Mx+q∈KandyTx=0.We say that M∈Q if SOLCP(M,q) has a solution for all q∈IRn. An n×n real matrix A is said to be a Z-matrix with respect to K iff:x∈K,y∈KandxTy=0 ⟹xTMy≤0.Let ΦM(q) denote the set of all solutions to SOLCP(M,q). The following results are shown in this paper:•If M∈Z∩Q, then ΦM is Lipschitz continuous if and only if M is positive definite on the boundary of K.•If M is symmetric, then ΦM is Lipschitz continuous if and only if M is positive definite.

Solution analysis for the pseudomonotone second-order cone linear complementarity problem

In this paper, we are concerned with the set of the solutions and the geometric property of the pseudomonotone second-order cone linear complementarity problems (SOCLCP). Based on Tao’s recent work [Tao, J. Optim. Theory Appl., 159, 41–56 (2013)] on pseudomonotone LCP on Euclidean Jordan algebras, we characterize the set of solutions and also derive intrinsic properties that reveal the underlying geometry of the pseudomonotone SOCLCP.

Feasibility and structural feature on monotone second-order cone linear complementarity problems in Hilbert space

Given a real finite-dimensional or infinite-dimensional Hilbert space H with a Jordan product, the second-order cone linear complementarity problem(SOCLCP)is considered. Some conditions are investigated, for which the SOCLCP is feasible and solvable for any element q∈H. The solution set of a monotone SOCLCP is also characterized. It is shown that the second-order cone and Jordan product are interconnected.

A Krylov Subspace Method for Large-Scale Second-Order Cone Linear Complementarity Problem

In this paper, we first show that the second-order cone linear complementarity problem (SOCLCP) can be solved by finding a positive zero $s_*\in \mathbb{R}$ of a particular rational function $h(s)$, and we then propose a Krylov subspace method to reduce $h(s)$ to $h_{\ell}(s)$ as in the model reduction. The zero $s_*$ of $h(s)$ can be accurately approximated by that of $h_{\ell}(s)=0$, which itself can be cast as a small eigenvalue problem. The new method is made possible by a complete characterization of the curve of $h(s)$, and it has several advantages over the bisection-Newton (\sf BN) iteration recently proposed by [L.-H. Zhang and W. H. Yang, Math. Comp., 83 (2013), pp. 1701--1720] and shown to be very efficient for small- to medium-size problems. The method is tested and compared against the \sf BN iteration and two other state-of-the-art packages: \sf SDPT3 and \sf SeDuMi. Our numerical results show that the method is very efficient for both small to medium dense problems and large-scale ones.

A power penalty method for second-order cone linear complementarity problems

We propose a power penalty method for solving the second-order cone linear complementarity problems (SOCLCPs), which is an extension of Wang and Yang’s research (Wang and Yang, 2008). Using this method, the SOCLCP is converted to asymptotic power penalty equations (PPEs). We prove that the solution sequence of the asymptotic PPEs converges to the solution of the SOCLCP at an exponential rate under a mild assumption. Numerical results are reported to examine the efficiency of the proposed method.

An Efficient Matrix Splitting Method for the Second-Order Cone Complementarity Problem

Given a symmetric and positive (semi)definite $n$-by-$n$ matrix $M$ and a vector, in this paper, we consider the matrix splitting method for solving the second-order cone linear complementarity problem (SOCLCP). The matrix splitting method is among the most widely used approaches for large scale and sparse classical linear complementarity problems, and its linear convergence is proved by [Luo and Tseng, SIAM J. Control Optim., 30 (1992), pp. 408--425]. Our first contribution is to prove that, when the general matrix splitting algorithm is applied to SOCLCP with $M$ symmetric and positive definite, it also converges at least linearly. Numerically, our second contribution is to propose a special and efficient matrix splitting algorithm, the block successive overrelaxation method, for solving the SOCLCP. The algorithm makes good use of the underlying geometry of the SOCLCP and each iteration only involves solving triangular linear systems and requires $O(n^2)$ flops; moreover, the algorithm does not destroy the sparse structure of $M$ and is able to exploit the sparsity effectively. Our code BSOR_BN_L based on this method is tested against four other state-of-the-art methods on problems with dense, ill-conditioned symmetric and positive definite matrices, as well as on problems with large scale sparse, symmetric and positive (semi)definite matrices. The numerical results are quite encouraging and BSOR_BN_L exhibits a very good performance in terms of its speed, accuracy, and efficiency in exploiting the sparsity of $M$.

An efficient algorithm for second-order cone linear complementarity problems

Recently, the globally uniquely solvable (GUS) property of the linear transformation M ∈ R n × n M\in \mathbb {R}^{n\times n} in the second-order cone linear complementarity problem (SOCLCP) receives much attention and has been studied substantially. Yang and Yuan contributed a new characterization of the GUS property of the linear transformation, which is formulated by basic linear-algebra-related properties. In this paper, we consider efficient numerical algorithms to solve the SOCLCP where the linear transformation M M has the GUS property. By closely relying on the new characterization of the GUS property, a globally convergent bisection method is developed in which each iteration can be implemented using only 2 n 2 2n^2 flops. Moreover, we also propose an efficient Newton method to accelerate the bisection algorithm. An attractive feature of this Newton method is that each iteration only requires 5 n 2 5n^2 flops and converges quadratically. These two approaches make good use of the special structure contained in the SOCLCP and can be effectively combined to yield a fast and efficient bisection-Newton method. Numerical testing is carried out and very encouraging computational experiments are reported.

The GUS-property of second-order cone linear complementarity problems

The globally uniquely solvable (GUS) property of the linear transformation of the linear complementarity problems over symmetric cones has been studied recently by Gowda et al. via the approach of Euclidean Jordan algebra. In this paper, we contribute a new approach to characterizing the GUS property of the linear transformation of the second-order cone linear complementarity problems (SOCLCP) via some basic linear algebra properties of the involved matrix of SOCLCP. Some more concrete and checkable sufficient and necessary conditions for the GUS property are thus derived.

New second-order cone linear complementarity formulation and semi-smooth Newton algorithm for finite element analysis of 3D frictional contact problem

We propose a new second-order cone linear complementarity problem (SOCLCP) formulation for the numerical finite element analysis of three-dimensional (3D) frictional contact problems by the parametric variational principle. Specifically, we develop a regularization technique to resolve the multi-valued difficulty involved in the frictional contact law, and use a second-order cone complementarity condition to handle the regularized Coulomb friction law in contact analysis. The governing equations of the 3D frictional contact problem is represented by an SOCLCP via the parametric variational principle and the finite element method, which avoids the polyhedral approximation to the Coulomb friction cone so that the problem to be solved has much smaller size and the solution has better accuracy. In this paper, we reformulate the SOCLCP as a semi-smooth system of equations via a one-parametric class of second-order cone complementarity functions, and then apply the non-smooth Newton method for solving this system. Numerical results confirm the effectiveness and robustness of the SOCLCP approach developed.

A class of polynomial interior-point algorithms for the Cartesian [formula omitted] second-order cone linear complementarity problem

In this paper a class of polynomial interior-point algorithms for the Cartesian P ∗ ( κ ) second-order cone linear complementarity problem based on a parametric kernel function, with parameters p ∈ [ 0 , 1 ] and q ≥ 1 , are presented. The proposed parametric kernel function is used both for determining the search directions and for measuring the distance between the given iterate and the μ -center for the algorithms. Moreover, the currently best known iteration bounds for the large- and small-update methods, namely, O ( ( 1 + 2 κ ) N log N log N ε ) and O ( ( 1 + 2 κ ) N log N ε ) , are obtained, respectively, which reduce the gap between the practical behavior of the algorithms and its theoretical performance results.