Abstract
In this paper, we comprehensively study optimality conditions for rank-constrained matrix optimization (RCMO). By calculating the Clarke tangent and normal cones to a rank-constrained set, along with the given Fréchet, Mordukhovich normal cones, we investigate four kinds of stationary points of the RCMO and analyze the relations between each stationary point and local/global minimizer of the RCMO. Furthermore, the second-order optimality condition of the RCMO is achieved with the help of the Clarke tangent cone.
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This research was supported by the National Natural Science Foundation of China (Nos. 11431002 and 11371116).
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Li, XR., Song, W. & Xiu, NH. Optimality Conditions for Rank-Constrained Matrix Optimization. J. Oper. Res. Soc. China 7, 285–301 (2019). https://doi.org/10.1007/s40305-019-00245-0
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DOI: https://doi.org/10.1007/s40305-019-00245-0
Keywords
- Matrix optimization
- Rank constraint
- Normal cone
- First-order optimality condition
- Second-order optimality condition