Abstract
Given a real Hilbert space H with a Jordan product and \({\Omega\subset H}\) being the Lorentz cone, \({q\in H}\), and let T : H → H be a bounded linear transformation, the corresponding linear complementarity problem is denoted by LCP(T, Ω, q). In this paper, we introduce the concepts of the column-sufficiency and row-sufficiency of T. In particular, we show that the row-sufficiency of T is equivalent to the existence of the solution of LCP(T, Ω, q) under an operator commutative condition; and that the column-sufficiency along with cross commutative property is equivalent to the convexity of the solution set of LCP(T, Ω, q). In our analysis, the properties of the Jordan product and the Lorentz cone in H are interconnected.
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This work was partially supported by National Nature Science Foundation of China (No. 10871144) and the Natural Science Foundation of Tianjin (No. 07JCYBJC05200).
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Miao, XH., Huang, ZH. The column-sufficiency and row-sufficiency of the linear transformation on Hilbert spaces. J Glob Optim 49, 109–123 (2011). https://doi.org/10.1007/s10898-010-9537-5
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DOI: https://doi.org/10.1007/s10898-010-9537-5