Matrix optimisation over low-rank spectral sets: Stationary points, local and global minimizers

Li, XinRong, Xiu, Naihua and Zhou, Shenglong (2020) Matrix optimisation over low-rank spectral sets: Stationary points, local and global minimizers. Journal of Optimization Theory and Applications, 184 (3), 895-930. (doi:10.1007/s10957-019-01606-8).

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Abstract

In this paper, we consider matrix optimization with the variable as a matrix that is constrained into a low-rank spectral set, where the low-rank spectral set is the intersection of a low-rank set and a spectral set. Three typical spectral sets are considered, yielding three low-rank spectral sets. For each low-rank spectral set, we first calculate the projection of a given point onto this set and the formula of its normal cone, based on which the induced stationary points of matrix optimization over low-rank spectral sets are then investigated. Finally, we reveal the relationship between each stationary point and each local/global minimizer.

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Submitted date: 2018

Accepted/In Press date: 23 October 2019

e-pub ahead of print date: 9 December 2019

Published date: 1 March 2020

Additional Information: Funding Information: This work was supported in part by the National Natural Science Foundation of China (11431002). Publisher Copyright: © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords: Global minimizer, Local minimizer, Low-rank spectral set, Matrix optimization, Stationary point