Symmetric Linear Complementarity Problem Research Articles

Interior-point algorithm for symmetric cone horizontal linear complementarity problems based on a new class of algebraically equivalent transformations

AbstractWe generalize a primal-dual interior-point algorithm (IPA) proposed recently in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) to $$P_*(\kappa )$$ P ∗ ( κ ) -horizontal linear complementarity problems (LCPs) over Cartesian product of symmetric cones. The algorithm is based on the algebraic equivalent transformation (AET) technique with a new class of AET functions. The new class is a modification of the class of AET functions proposed in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) where only two conditions are used as opposed to three used in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022). Furthermore, the algorithm is a feasible algorithm that uses full Nesterov-Todd steps, hence, no calculation of step-size is necessary. Nevertheless, we prove that the proposed IPA has the iteration bound that matches the best known iteration bound for IPAs solving these types of problems.

New Predictor–Corrector Algorithm for Symmetric Cone Horizontal Linear Complementarity Problems

We propose a new predictor–corrector interior-point algorithm for solving Cartesian symmetric cone horizontal linear complementarity problems, which is not based on a usual barrier function. We generalize the predictor–corrector algorithm introduced in Darvay et al. (SIAM J Optim 30:2628–2658, 2020) to horizontal linear complementarity problems on a Cartesian product of symmetric cones. We apply the algebraically equivalent transformation technique proposed by Darvay (Adv Model Optim 5:51–92, 2003), and we use the difference of the identity and the square root function to determine the new search directions. In each iteration, the proposed algorithm performs one predictor and one corrector step. We prove that the predictor–corrector interior-point algorithm has the same complexity bound as the best known interior-point methods for solving these types of problems. Furthermore, we provide a condition related to the proximity and update parameters for which the introduced predictor-corrector algorithm is well defined.

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.

A rigorous and efficient explicit algorithm for irreversibility enforcement in phase-field finite element modeling of brittle crack propagation

In the present work, a computationally efficient and explicit algorithm for the rigorous enforcement of the irreversibility constraint in the phase-field modeling of brittle fracture is presented. The proposed approach is staggered and relies on the alternate minimization of the total energy functional. The phase-field evolution turns out to be governed by a complementarity boundary-value problem, where the complementarity stems from the irreversibility, while the boundary-value problem is originated by the presence of the gradient term in the phase-field functional. Several different techniques have been proposed in the literature to account for damage irreversibility in a computationally effective way. Following a similar approach proposed in the past for a gradient-plasticity problem, a particularly simple and effective solution strategy based on the Projected Successive Over-Relaxation (PSOR) method for constrained optimization, where an iterative explicit scheme is used for the solution of symmetric linear complementarity problems, is presented. Even though the proposed method is restricted to linear complementarity problems, it can be applied to the numerical simulation of a wide range of problems discussed in the literature. The performance of the suggested solution algorithm on a number of test cases is compared with that of a recently proposed penalty approach for the approximated enforcement of irreversibility.

Full Nesterov-Todd step feasible interior-point algorithm for symmetric cone horizontal linear complementarity problem based on a positive-asymptotic barrier function

We present a feasible full step interior-point algorithm to solve the horizontal linear complementarity problem defined on a Cartesian product of symmetric cones, which is not based on a usual barrier function. The full steps are scaled utilizing the Nesterov-Todd (NT) scaling point. Our approach generates the search directions leading to the full-NT steps by algebraically transforming the centring equation of the system which defines the central trajectory using the induced barrier of a so-called positive-asymptotic kernel function. We establish the global convergence as well as a local quadratic rate of convergence of our proposed method. Finally, we demonstrate that our algorithm bears a complexity bound matching the best available one for the algorithms of its kind.

A Full-NT Step Infeasible Interior-Point Algorithm for Mixed Symmetric Cone LCPs

An infeasible interior-point algorithm for mixed symmetric cone linear complementarity problems is proposed. Using the machinery of Euclidean Jordan algebras and Nesterov-Todd search direction, the convergence analysis of the algorithm is shown and proved. Moreover, we obtain a polynomial time complexity bound which matches the currently best known iteration bound for infeasible interior-point methods.

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.

Multisplitting Iterative Methods with General Weighting Matrices for Solving Symmetric Positive Linear Complementarity Problem

In this paper, by making use of optimal models, we study the weighting matrices of the multisplitting parallel methods for solving the symmetric positive definite linear complementarity problem, which is a powerful alternative for solving the large sparse linear complementarity problems. In our multisplitting there is only one that is required to be P-regular splitting and all the others can be constructed arbitrarily, which not only decreases the difficulty of constructing the multisplitting of the coefficient matrix, but also relaxes the constraints to the weighting matrices (unlike the standard methods, they are not necessarily nonnegative diagonal scalar matrices or given in advance). Finally, we prove the convergence of this new method.

A predictor-corrector infeasible-interior-point method for the Cartesian -LCP over symmetric cones with iteration complexity

In this paper, we present a predictor-corrector infeasible-interior-point method for symmetric cone linear complementarity problem (SCLCP) with the Cartesian -property (-SCLCP). This method is based on a wide neighbourhood, which is an even wider neighbourhood than the negative infinity neighbourhood. We show that the iteration-complexity bound of the proposed algorithm for a commutative class of search directions is , where is the condition number of matrix G, is the handicap of the problem, r is the rank of the associated Euclidean Jordan algebra and is a given tolerance. To our knowledge, this is the best complexity result obtained so far for the wide neighbourhood infeasible-interior-point methods for the Cartesian -SCLCPs.

A Lipschitzian error bound for monotone symmetric cone linear complementarity problem

We first discuss some properties of the solution set of a monotone symmetric cone linear complementarity problem (SCLCP), and then consider the limiting behaviour of a sequence of strictly feasible solutions within a wide neighbourhood of central trajectory for the monotone SCLCP. Under assumptions of strict complementarity and Slater’s condition, we provide four different characterizations of a Lipschitzian error bound for the monotone SCLCP in general Euclidean Jordan algebras. Thanks to the observation that a pair of primal-dual convex quadratic symmetric cone programming (CQSCP) problems can be exactly formulated as the monotone SCLCP, thus we obtain the same error bound results for CQSCP as a by-product.

MSSOR-based alternating direction method for symmetric positive-definite linear complementarity problems

Finding the solution of linear complementarity problem (LCP) is an important subject in scientific computing, engineering and economic applications. In the literature, many iterative methods have been proposed, especially, when the matrix of the problem is a real positive definite or an H +? matrix. In this paper, we propose a new method using an alternating direction (AD) technique based on a modified symmetric successive overrelaxation (MSSOR) for LCP, assuming that the real matrix of the LCP is symmetric positive definite. This approach is an inner/outer iteration method, in which AD method is used as an outer iteration, and the MSSOR method as an inner one. We provide a detailed proof for the convergence of the proposed method. In addition, numerical results show that the proposed method is feasible and effective.

An infeasible interior-point algorithm based on modified Nesterov and Todd directions for symmetric linear complementarity problem

We present an infeasible interior-point algorithm for symmetric linear complementarity problem based on modified Nesterov–Todd directions by using Euclidean Jordan algebras. The algorithm decreases the duality gap and the feasibility residual at the same rate. In this algorithm, we construct strictly feasible iterates for a sequence of perturbations of the given problem. Each main iteration of the algorithm consists of a feasibility step and a number of centring steps. The starting point in the first iteration is strictly feasible for a perturbed problem. The feasibility steps lead to a strictly feasible iterate for the next perturbed problem. By using centring steps for the new perturbed problem, a strictly feasible iterate is obtained to be close to the central path of the new perturbed problem. Furthermore, giving a complexity analysis of the algorithm, we derive the currently best-known iteration bound for infeasible interior-point methods.

A weighted-path-following method for symmetric cone linear complementarity problems

In this paper a weighted-path-following interior-point algorithm for linear complementarity problem over symmetric cones is proposed that uses new search directions. The complexity results of the new algorithm derived and proved that the proposed algorithm has quadratically convergent with polynomial-time. We conclude that following the central path yields to the best iteration bound in this case as well.

A Generic Interior-point Algorithm for Monotone Symmetric Cone Linear Complementarity Problems Based on a New Kernel Function

Kernel functions play an important role in defining new search directions for interior-point algorithms for solving monotone linear complementarity problems. In this paper we present a new kernel function which yields the complexity bounds \({\mathcal O}(\sqrt{r}\log r\log\frac{r}{\epsilon})\) and \({\mathcal O}(\sqrt{r}\log\frac{r}{\epsilon})\) for large-and small-update methods, respectively, which are currently the best known bounds for such methods.

A Smoothing Newton Algorithm for a Class of Non-monotonic Symmetric Cone Linear Complementarity Problems

Recently, the study of symmetric cone complementarity problems has been a hot topic in the literature. Many numerical methods have been proposed for solving such a class of problems. Among them, the problems concerned are generally monotonic. In this paper, we consider symmetric cone linear complementarity problems with a class of non-monotonic transformations. A smoothing Newton algorithm is extended to solve this class of non-monotonic symmetric cone linear complementarity problems; and the algorithm is proved to be well-defined. In particular, we show that the algorithm is globally and locally quadratically convergent under mild assumptions. The preliminary numerical results are also reported.

A new interior-point algorithm based on modified Nesterov–Todd direction for symmetric cone linear complementarity problem

We present a new primal-dual path-following interior-point algorithm for linear complementarity problem over symmetric cones. The algorithm is based on a reformulation of the central path for finding the search directions. For a full Nesterov–Todd step feasible interior-point algorithm based on the search directions, the complexity bound of the algorithm with the small update approach is the best available.

Subspace Accelerated Matrix Splitting Algorithms for Asymmetric and Symmetric Linear Complementarity Problems

This paper studies the solution of both asymmetric and symmetric linear complementarity problems by two-phase methods that consist of an active set prediction phase and an acceleration phase. The prediction phase employs matrix splitting iterations that are tailored to the structure of the linear complementarity problems studied in this paper. In the asymmetric case, the task of pairing an acceleration phase with matrix splitting iterations is achieved by exploiting a contraction property associated with certain matrix splittings. For symmetric problems, a similar task is achieved by utilizing decent properties of specific matrix splitting iterations and projected searches. The superior optimal active set identification property of matrix splitting iterations is illustrated with numerical experiments, which also demonstrate the general efficiency of the proposed methods.

Multiple birth support vector machine for multi-class classification

For multi-class classification problem, a novel algorithm, called as multiple birth support vector machine (MBSVM), is proposed, which can be considered as an extension of twin support vector machine. Our MBSVM has been compared with the several typical support vector machines. From theoretical point of view, it has been shown that its computational complexity is remarkably low, especially when the class number K is large. Based on our MBSVM, the dual problems of MBSVM are equivalent to symmetric mixed linear complementarity problems to which successive overrelaxation (SOR) can be directly applied. We establish our SOR algorithm for MBSVM. The SOR algorithm handles one data point at a time, so it can process large dataset that need no reside in memory. From practical point of view, its accuracy has been validated by the preliminary numerical experiments.

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.

Smoothing Newton methods for symmetric cone linear complementarity problem with the Cartesian $P$/$P_0$-property

A smoothing Newton method based on the CHKS smoothing function fora class of non-monotone symmetric cone linear complementarity problem (SCLCP) with the Cartesian $P$-propertyand a regularization smoothing Newton method for SCLCP with the Cartesian $P_0$-property are proposed.The existence of Newton directions and the boundedness of iterates, two important theoretical issues encountered in the smoothing method, are showed for these two classes of non-monotone SCLCP. Finally, we show that these two algorithms are globally convergent. Moreover, they are globally linear and locally quadratic convergent under a nonsingular assumption.