Properties associated with the epigraph of the $$l_1$$ l 1 norm function of projection onto the nonnegative orthant

Yong-Jin Liu () and Li Wang ()
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Yong-Jin Liu: Shenyang Aerospace University
Li Wang: Shenyang Aerospace University

Mathematical Methods of Operations Research, 2016, vol. 84, issue 1, No 9, 205-221

Abstract: Abstract This paper studies some properties associated with a closed convex cone $$\mathcal {K}_{1+}$$ K 1 + , which is defined as the epigraph of the $$l_1$$ l 1 norm function of the metric projection onto the nonnegative orthant. Specifically, this paper presents some properties on variational geometry of $$\mathcal {K}_{1+}$$ K 1 + such as the dual cone, the tangent cone, the normal cone, the critical cone and its convex hull, and others; as well as the differential properties of the metric projection onto $$\mathcal {K}_{1+}$$ K 1 + including the directional derivative, the B-subdifferential, and the Clarke’s generalized Jacobian. These results presented in this paper lay a foundation for future work on sensitivity and stability analysis of the optimization problems over $$\mathcal {K}_{1+}$$ K 1 + .

Keywords: Metric projection; Epigraph; Variational geometry; Differential property; 90C25; 90C30; 65K05 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s00186-016-0540-6

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