14,793 research outputs found
Inequalities for selected eigenvalues of the product of matrices
The product of a Hermitian matrix and a positive semidefinite matrix has only
real eigenvalues. We present bounds for sums of eigenvalues of such a product.Comment: to appear in AMS Proceeding
A semi-proximal-based strictly contractive Peaceman-Rachford splitting method
The Peaceman-Rachford splitting method is very efficient for minimizing sum
of two functions each depends on its variable, and the constraint is a linear
equality. However, its convergence was not guaranteed without extra
requirements. Very recently, He et al. (SIAM J. Optim. 24: 1011 - 1040, 2014)
proved the convergence of a strictly contractive Peaceman-Rachford splitting
method by employing a suitable underdetermined relaxation factor. In this
paper, we further extend the so-called strictly contractive Peaceman-Rachford
splitting method by using two different relaxation factors, and to make the
method more flexible, we introduce semi-proximal terms to the subproblems. We
characterize the relation of these two factors, and show that one factor is
always underdetermined while the other one is allowed to be larger than 1. Such
a flexible conditions makes it possible to cover the Glowinski's ADMM whith
larger stepsize. We show that the proposed modified strictly contractive
Peaceman-Rachford splitting method is convergent and also prove
convergence rate in ergodic and nonergodic sense, respectively. The numerical
tests on an extensive collection of problems demonstrate the efficiency of the
proposed method
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