On degeneracy in linear complementarity problems

AbstractLet M be an n×n matrix and q an nth order vector. Then the linear complementarity problem LCP(q, M) is defined as follows: determine x⩾0 such that w=Mx+q⩾0 and xTw=0. A vector x which satisfies these conditions is called a solution of the problem, and a solution for which xi=wi=0 for at least one value of i is termed degenerate. If the solutions of LCP(q, M) are nondegenerate and their number is odd (even), we say that the solution set has odd (even) parity, and Murty has shown that this parity is determined uniquely by M. In this paper the idea of parity is extended to degenerate solutions and, through these, to solution sets containing both degenerate and nondegenerate solutions. These results are then used to give a generalization of Lemke's method and to analyse the stability of certain degenerate solutions of linear complementarity problems

Implementation of different computational variations of biconjugate residual methods

AbstractIn this paper, we describe the derivation of the biconjugate residual (BCR) method from the general framework of the Block-CG algorithm; we then introduce different versions of BCR and test their numerical performance

On the ultimate convergence rates for isotropic algorithms and the best choices among various forms of isotropy

In this paper, we show universal lower bounds for isotropic algorithms, that hold for any algorithm such that each new point is the sum of one already visited p oint plus one random isotropic direction multiplied by any step size (whenever the step size is chosen by an oracle with arbitrarily high computational power). The bound is 1 − O(1/d) for the constant in the linear convergence (i.e. the constant C such that the distance to the optimum after n steps is upp er b ounded by C n ), as already seen for some families of evolution strategies in [19, 12], in contrast with 1 − O(1) for the reverse case of a random step size and a direction chosen by an oracle with arbitrary high computational power. We then recall that isotropy does not uniquely determine the distribution of a sample on the sphere and show that the convergence rate in isotropic algorithms is improved by using stratified or antithetic isotropy instead of naive isotropy. We show at the end of the pap er that b eyond the mathematical proof, the result holds on exp eriments. We conclude that one should use antithetic-isotropy or stratified-isotropy, and never standard-isotropy

Finite reduction and Morse index estimates for mechanical systems

A simple version of exact finite dimensional reduction for the variational setting of mechanical systems is presented. It is worked out by means of a thorough global version of the implicit function theorem for monotone operators. Moreover, the Hessian of the reduced function preserves all the relevant information of the original one, by Schur's complement, which spontaneously appears in this context. Finally, the results are straightforwardly extended to the case of a Dirichlet problem on a bounded domain.Comment: 13 pages; v2: minor changes, to appear in Nonlinear Differential Equations and Application

Ab Initio Evidence for the Formation of Impurity d(3z^2-r^2) Holes in Doped La_{2-x}Sr_xCuO_4

Using the spin unrestricted Becke-3-Lee-Yang-Parr density functional, we computed the electronic structure of explicitly doped La_{2-x}Sr_xCuO_4 (x = 0.125, 0.25, and 0.5). At each doping level, an impurity hole band is formed within the undoped insulating gap. This band is well-localized to CuO_6 octahedra adjacent to the Sr impurities. The nature of the impurity hole is A_{1g} in symmetry, formed primarily from the z^2 orbital on the Cu and p_z orbitals on the apical O's. There is a strong triplet coupling of this hole with the intrinsic B_{1g} Cu x^2-y^2/O1 p_{sigma} hole on the same site. Optimization of the c coordinate of the apical O's in the doped CuO_6 octahedron lead to an asymmetric anti-Jahn-Teller distortion of the O2 atoms toward the central Cu. In particular, the O2 atom between the Cu and Sr is displaced 0.26 A while the O2 atom between the Cu and La is displaced 0.10 A. Contrary to expectations, investigation of a 0.1 A enhanced Jahn-Teller distortion of this octahedron does not force formation of an x^2-y^2 hole, but instead leads to migration of the z^2 hole to the four other CuO_6 octahedra surrounding the Sr impurity. This latter observation offers a simple explanation for the bifurcation of the Sr-O2 distance revealed in x-ray absorption fine structure data.Comment: Submitted to Phys. Rev. B. See http://www.firstprinciples.com for more informatio