Abstract
This paper generalizes the factorization theorem of Gouveia, Parrilo and Thomas to a broader class of convex sets. Given a general convex set, the authors define a slack operator associated to the set and its polar according to whether the convex set is full dimensional, whether it is a translated cone and whether it contains lines. The authors strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone, but also its recession cone is the image of the linear slice of the closed convex cone. The authors show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.
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This paper was supported by Equipment Pre-Research Field Fund under Grant Nos. JZX7Y20190258055501, JZX7Y20190243016801, the National Natural Science Foundation of China under Grant No. 11901544, the National Key Research Project of China under Grant No. 2018YFA0306702 and the National Natural Science Foundation of China under Grant No. 11571350. This work is also supported by National Institute for Mathematical Sciences 2014 Thematic Program on Applied Algebraic Geometry in Daejeon, South Korea.
This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.
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Wang, C., Zhi, L. Lifts of Non-Compact Convex Sets and Cone Factorizations. J Syst Sci Complex 33, 1632–1655 (2020). https://doi.org/10.1007/s11424-020-9050-y
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DOI: https://doi.org/10.1007/s11424-020-9050-y