Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities

Dezhou Kong (), Lishan Liu () and Yonghong Wu ()
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Dezhou Kong: Shandong Agricultural University
Lishan Liu: Qufu Normal University
Yonghong Wu: Curtin University

Journal of Optimization Theory and Applications, 2017, vol. 173, issue 1, No 6, 117-130

Abstract: Abstract In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others.

Keywords: Lorentz cone; Variational inequality; Metric projection; Complementarity problem; Quasi-lattice; 47H07; 39B62; 47J20; 47H10; 49J40 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10957-017-1084-5

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