The eigenvalue complementarity problem

In this paper an eigenvalue complementarity problem (EiCP) is studied, which finds its origins in the solution of a contact problem in mechanics. The EiCP is shown to be equivalent to a Nonlinear Complementarity Problem, a Mathematical Programming Problem with Complementarity Constraints and a Global Optimization Problem. A finite Reformulation–Linearization Technique (Rlt)-based tree search algorithm is introduced for processing the EiCP via the lattermost of these formulations. Computational experience is included to highlight the efficacy of the above formulations and corresponding techniques for the solution of the EiCP.     

On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm

This paper is devoted to the eigenvalue complementarity problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differentiable optimization program involving the Rayleigh quotient on a simplex (Queiroz et al., Math. Comput. 73, 1849–1863, 2004). We discuss a logarithmic function and a quadratic programming formulation to find a complementarity eigenvalue by computing a stationary point of an appropriate merit function on a special convex set. A variant of the spectral projected gradient algorithm with a specially designed line search is introduced to solve the EiCP. Computational experience shows that the application of this algorithm to the logarithmic function formulation is a quite efficient way to find a solution to the symmetric EiCP.     

The inverse eigenvalue problem for symmetric quasi anti-bidiagonal matrices

In this paper we construct the symmetric quasi anti-bidiagonal matrix that its eigenvalues are given, and show that the problem is also equivalent to the inverse eigenvalue problem for a certain symmetric tridiagonal matrix which has the same eigenvalues. Not only elements of the tridiagonal matrix come from quasi anti-bidiagonal matrix, but also the places of elements exchange based on some conditions.     

An eigenvalue problem for a symmetric Toeplitz matrix

An algorithm is developed which determines eigenvalues for a symmetric Toeplitz matrix. To this end, we substantiate the generality of eigenvalues problems for a symmetric Toeplitz matrix and for a persymmetric Hankel one. The latter is reduced to an eigenvalue problem for a persymmetric Jacobi matrix. In the even order case, the problem reduces to a Jacobi matrix with halved order.     

An algorithm for the symmetric generalized eigenvalue problem

A new method is presented for the solution of the matrix eigenvalue problem Ax=λBx, where A and B are real symmetric square matrices and B is positive semidefinite. It reduces A and B to diagonal form by congruence transformations that preserve the symmetry of the problem. This method is closely related to the QR algorithm for real symmetric matrices.     

Sort-Jacobi and generalizations of the symmetric eigenvalue problem

In this paper, it is sketched how the Sort-Jacobi method for the symmetric eigenvalue problem extends to the (–1)-eigenspace of the Cartan involution on an arbitrary semisimple Lie algebra. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well-known normal form problems such as the symmetric, Hermitian, skewsymmetric, symmetric and skew-symmetric ℝ-Hamiltonian eigenvalue problem and the singular value decomposition. It allows a unified treatment of the above eigenvalue methods, including local convergence analysis. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bisection acceleration for the symmetric tridiagonal eigenvalue problem

We present new algorithms that accelerate the bisection method for the symmetric tridiagonal eigenvalue problem. The algorithms rely on some new techniques, including a new variant of Newton's iteration that reaches cubic convergence (right from the start) to the well separated eigenvalues and can be further applied to acceleration of some other iterative processes, in particular, of the divide-and-conquer methods for the symmetric tridiagonal eigenvalue problem. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the numerical solution of the quadratic eigenvalue complementarity problem

The Quadratic Eigenvalue Complementarity Problem (QEiCP) is an extension of the Eigenvalue Complementarity Problem (EiCP) that has been introduced recently. Similar to the EiCP, the QEiCP always has a solution under reasonable hypotheses on the matrices included in its definition. This has been established in a previous paper by reducing a QEiCP of dimension n to a special 2n-order EiCP. In this paper we propose an enumerative algorithm for solving the QEiCP by exploiting this equivalence with an EiCP. The algorithm seeks a global minimum of a special Nonlinear Programming Problem (NLP) with a known global optimal value. The algorithm is shown to perform very well in practice but in some cases terminates with only an approximate optimal solution to NLP. Hence, we propose a hybrid method that combines the enumerative method with a fast and local semi-smooth method to overcome the latter drawback. This algorithm is also shown to be useful for computing a positive eigenvalue for an EiCP under similar assumptions. Computational experience is reported to demonstrate the efficacy and efficiency of the hybrid enumerative method for solving the QEiCP.     

Power iteration and inverse power iteration for eigenvalue complementarity problem

In this paper, an inverse complementarity power iteration method (ICPIM) for solving eigenvalue complementarity problems (EiCPs) is proposed. Previously, the complementarity power iteration method (CPIM) for solving EiCPs was designed based on the projection onto the convex cone K. In the new algorithm, a strongly monotone linear complementarity problem over the convex cone K is needed to be solved at each iteration. It is shown that, for the symmetric EiCPs, the CPIM can be interpreted as the well‐known conditional gradient method, which requires only linear optimization steps over a well‐suited domain. Moreover, the ICPIM is closely related to the successive quadratic programming (SQP) via renormalization of iterates. The global convergence of these two algorithms is established by defining two nonnegative merit functions with zero global minimum on the solution set of the symmetric EiCP. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithms.     

On the computation of all eigenvalues for the eigenvalue complementarity problem

In this paper, a parametric algorithm is introduced for computing all eigenvalues for two Eigenvalue Complementarity Problems discussed in the literature. The algorithm searches a finite number of nested intervals \([\bar{l}, \bar{u}]\) in such a way that, in each iteration, either an eigenvalue is computed in \([\bar{l}, \bar{u}]\) or a certificate of nonexistence of an eigenvalue in \([\bar{l}, \bar{u}]\) is provided. A hybrid method that combines an enumerative method [1] and a semi-smooth algorithm [2] is discussed for dealing with the Eigenvalue Complementarity Problem over an interval \([\bar{l}, \bar{u}]\) . Computational experience is presented to illustrate the efficacy and efficiency of the proposed techniques.     

Tensor eigenvalue complementarity problems

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre’s hierarchy of semidefinite relaxations. We show that it has finite convergence for generic tensors. Numerical experiments are presented to show the efficiency of proposed methods.     

Extremal inverse eigenvalue problem for symmetric doubly arrow matrices

In this paper, we consider the following inverse eigenvalue problem: to construct a real symmetric doubly arrow matrix A from the minimal and maximal eigenvalues of all its leading principal submatrices. The necessary and sufficient condition for the solvability of the problem is derived. We also give a necessary and sufficient condition in order that the constructed matrices can be nonnegative. Our results are constructive and they generate algorithmic procedures to construct such matrices.     

An algorithm for the generalized symmetric tridiagonal eigenvalue problem

In this paper we present an algorithm, parallel in nature, for finding eigenvalues of a symmetric definite tridiagonal matrix pencil. Our algorithm employs the determinant evaluation, split-and-merge strategy and Laguerre's iteration. Numerical results on both single and multiprocessor computers are presented which show that our algorithm is reliable, efficient and accurate. It also enjoys flexibility in evaluating a partial spectrum.This research was supported in part by the Research Grant from University of West Florida.This research was supported in part by NSF under Grant CCR-9024840.     

The complementarity problem

For a given mapF from then-dimensional Euclidean spaceE ⁿ into itself, we consider the problem of finding a nonnegative vectorx inE ⁿ whose imageF(x) is also nonnegative and such that the two vectors are orthogonal. This problem is refered to in the literature as thecomplemcntarity problcm. The importance of the complementarity problem lies in the fact that it is the unifying mathematical form for a wide range of problems arising in different fields such as mathematical programming, game theory, economics, mechanics, etc This paper is concerned mainly with the question of the existence of a solution. Several existence theorems are given under various conditions on the mapF. These theorems cover the cases whenF is nonlinear nondifferentiable, nonlinear but differentiable, and affine.This paper was presented at the 7th Mathematical Programming Symposium, The Hague, The Netherlands.     

A projection and contraction method for symmetric cone complementarity problem

ABSTRACT

This paper considers the symmetric cone complementarity problem. A new projection and contraction method is presented which only requires some projection calculations and functional computations. It is proved that the iteration sequence produced by the proposed method converges to a solution of the symmetric cone complementarity problem under the condition that the underlying transformation is monotone. Numerical experiments also show the effectiveness of this method.     

Asynchronous parallel successive overrelaxation for the symmetric linear complementarity problem

Convergence is established for asynchronous parallel successive overrelaxation (SOR) algorithms for the symmetric linear complementarity problem. For the case of a strictly diagonally dominant matrix convergence is achieved for a relaxation factor interval of (0, 2] with line search, and (0, 1] without line search. Computational tests on the Sequent Symmetry S81 multiprocessor give speedup efficiency in the 43%–91% range for the cases for which convergence is established. The tests also show superiority of the asynchronous SOR algorithms over their synchronous counterparts.This material is based on research supported by National Science Foundation Grants DCR-8420963 and DCR-8521228 and Air Force Office of Scientific Research Grant AFOSR-86-0172.     

Two-stage parallel iterative methods for the symmetric linear complementarity problem

In this paper, we propose a two-stage parallel iterative method for solving the symmetric linear complementarity problem. When implemented in a parallel computing environment, the method decomposes the problem into subproblems which are solved by certain iterative procedures concurrently on separate processors. Convergence of the overall method is established under some mild assumptions on how the inner iterations are terminated. Applications of the proposed method to solve strictly convex quadratic programs are discused and numerical results on both a sequential computer (IBM 4381) and a super-computer (CRAYX-MP/24) are reported.This research was based on work supported by the National Science Foundation under grant ECS-8407240 and by a 1986 University Research and Development grant from Cray Research Inc.