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

Author

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  • Yong-Jin Liu

    (Shenyang Aerospace University)

  • Li Wang

    (Shenyang Aerospace University)

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 + .

Suggested Citation

  • Yong-Jin Liu & Li Wang, 2016. "Properties associated with the epigraph of the $$l_1$$ l 1 norm function of projection onto the nonnegative orthant," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 205-221, August.
    • Handle: RePEc:spr:mathme:v:84:y:2016:i:1:d:10.1007_s00186-016-0540-6
    DOI: 10.1007/s00186-016-0540-6
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    1. Jong-Shi Pang & Defeng Sun & Jie Sun, 2003. "Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 39-63, February.
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