4,512 research outputs found
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
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 cone eigenvalue complementarity problem for higher-order tensors
In this paper, we consider the tensor generalized eigenvalue complementarity
problem (TGEiCP), which is an interesting generalization of matrix eigenvalue
complementarity problem (EiCP). First, we given an affirmative result showing
that TGEiCP is solvable and has at least one solution under some reasonable
assumptions. Then, we introduce two optimization reformulations of TGEiCP,
thereby beneficially establishing an upper bound of cone eigenvalues of
tensors. Moreover, some new results concerning the bounds of number of
eigenvalues of TGEiCP further enrich the theory of TGEiCP. Last but not least,
an implementable projection algorithm for solving TGEiCP is also developed for
the problem under consideration. As an illustration of our theoretical results,
preliminary computational results are reported.Comment: 26 pages, 2 figures, 3 table
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