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

Abstract

This paper studies some properties associated with a closed convex cone $\mathcal {K}_{1+}$, which is defined as the epigraph of the $l_1$ norm function of the metric projection onto the nonnegative orthant. Specifically, this paper presents some properties on variational geometry of $\mathcal {K}_{1+}$ 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 $\mathcal {K}_{1+}$ 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 $\mathcal {K}_{1+}$.