On the Quadratic Eigenvalue Complementarity Problem

Abstract We introduce several new results on the Quadratic Eigenvalue Complementarity Problem (QEiCP), focusing on the nonsymmetric case, i,e, without making symmetry assumptions on the matrices defining the problem. First we establish a new sufficient condition for existence of solutions of this problem, which is somewhat more manageable than previously existent ones. This condition works through the introduction of auxiliary variables which leads to the reduction of QEiCP to an Eigenvalue Complementarity Problem (EiCP) in higher dimension. Hence, this reduction suggests a new strategy for solving QEiCP, which is also analyzed in the paper. We also present an upper bound for the number of solutions of QEiCP and exhibit some examples of instances of QEiCP whose solution set has large cardinality, without attaining though the just mentioned upper bound. We also investigate the numerical solution of the QEiCP by exploiting a nonlinear programming and a variational inequality formulations of QEiCP. Some numerical experiments are reported and illustrate the benefits and drawbacks of using these formulations for solving the QEiCP in practice

An Accelerated DC Programming Approach with Exact Line Search for The Symmetric Eigenvalue Complementarity Problem

In this paper, we are interested in developing an accelerated Difference-of-Convex (DC) programming algorithm based on the exact line search for efficiently solving the Symmetric Eigenvalue Complementarity Problem (SEiCP) and Symmetric Quadratic Eigenvalue Complementarity Problem (SQEiCP). We first proved that any SEiCP is equivalent to SEiCP with symmetric positive definite matrices only. Then, we established DC programming formulations for two equivalent formulations of SEiCP (namely, the logarithmic formulation and the quadratic formulation), and proposed the accelerated DC algorithm (BDCA) by combining the classical DCA with inexpensive exact line search by finding real roots of a binomial for acceleration. We demonstrated the equivalence between SQEiCP and SEiCP, and extended BDCA to SQEiCP. Numerical simulations of the proposed BDCA and DCA against KNITRO, FILTERED and MATLAB FMINCON for SEiCP and SQEiCP on both synthetic datasets and Matrix Market NEP Repository are reported. BDCA demonstrated dramatic acceleration to the convergence of DCA to get better numerical solutions, and outperformed KNITRO, FILTERED, and FMINCON solvers in terms of the average CPU time and average solution precision, especially for large-scale cases.Comment: 24 page

Copositivity and constrained fractional quadratic programs

Abstract We provide Completely Positive and Copositive Optimization formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed

Galois Unitaries, Mutually Unbiased Bases, and MUB-balanced states

A Galois unitary is a generalization of the notion of anti-unitary operators. They act only on those vectors in Hilbert space whose entries belong to some chosen number field. For Mutually Unbiased Bases the relevant number field is a cyclotomic field. By including Galois unitaries we are able to remove a mismatch between the finite projective group acting on the bases on the one hand, and the set of those permutations of the bases that can be implemented as transformations in Hilbert space on the other hand. In particular we show that there exist transformations that cycle through all the bases in every dimension which is an odd power of an odd prime. (For even primes unitary MUB-cyclers exist.) These transformations have eigenvectors, which are MUB-balanced states (i.e. rotationally symmetric states in the original terminology of Wootters and Sussman) if and only if d = 3 modulo 4. We conjecture that this construction yields all such states in odd prime power dimension.Comment: 32 pages, 2 figures, AMS Latex. Version 2: minor improvements plus a few additional reference

A Semismooth Newton Method for Tensor Eigenvalue Complementarity Problem

In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCP-function, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly semismooth but not differentiable, in which case the classical smoothing methods cannot apply. Furthermore, we propose a damped semismooth Newton method for tensor eigenvalue complementarity problem. A new procedure to evaluate an element of the generalized Jocobian is given, which turns out to be an element of the B-subdifferential under mild assumptions. As a result, the convergence of the damped semismooth Newton method is guaranteed by existing results. The numerical experiments also show that our method is efficient and promising

On the spherical convexity of quadratic functions

In this paper we study the spherical convexity of quadratic functions on spherically convex sets. In particular, conditions characterizing the spherical convexity of quadratic functions on spherical convex sets associated to the positive orthants and Lorentz cones are given