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Matrix Optimization Over Low-Rank Spectral Sets: Stationary Points and Local and Global Minimizers

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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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Notes

  1. A set \(\mathcal {K}\in \mathbb {R}^n\) is said to be symmetric if \(Px\in \mathcal {K}\) for every \(x\in \mathcal {K}\) and every \(P\in \mathbb {P}^n\), where \(\mathbb {P}^n\) denotes the set of all \(n\times n\) permutation matrices. ( For those matrices, that have only one nonzero entry in every row and column, which is 1, see [38].)

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (11431002).

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Authors and Affiliations

  1. Department of Applied Mathematics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Xinrong Li & Naihua Xiu

  2. School of Mathematics, University of Southampton, Southampton, UK

    Shenglong Zhou

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  1. Xinrong Li
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  2. Naihua Xiu
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  3. Shenglong Zhou
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Correspondence to Xinrong Li.

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Communicated by Suliman Saleh Al-Homidan.

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Li, X., Xiu, N. & Zhou, S. Matrix Optimization Over Low-Rank Spectral Sets: Stationary Points and Local and Global Minimizers. J Optim Theory Appl 184, 895–930 (2020). https://doi.org/10.1007/s10957-019-01606-8

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