The Geometric Properties of a Class of Nonsymmetric Cones
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DOI: 10.1007/s13226-020-0445-1
THE GEOMETRIC PROPERTIES OF A CLASS OF NONSYMMETRIC CONES Shiyun Wang Department of Science, Shenyang Institute of Aeronautical Engineering, Shenyang 110136, People’s Republic of China e-mail: [email protected] (Received 1 January 2019; after final revision 12 May 2019; accepted 18 May 2019) Geometric methods are important for researching the differential properties of metric projectors, sensitivity analysis, and the augmented Lagrangian algorithm. Sun [3] researched the relationship among the strong second-order sufficient condition, constraint nondegeneracy, B-subdifferential nonsingularity of the KKT system, and the strong regularity of KKT points in investigating nonlinear semidefinite programming problems. Geometric properties of cones are necessary in studying second-order sufficient condition and constraint nondegeneracy. In this paper, we study the geometric properties of a class of nonsymmetric cones, which is widely applied in optimization problems subjected to the epigraph of vector k-norm functions and low-rank-matrix approximations. We compute the polar, the tangent cone, the linear space of the tangent cone, the critical cone, and the affine hull of this critical cone. This paper will support future research into the sensitivity and algorithms of related optimization problems. Key words : Critical cone; geometric properties; nonsymmetric cone; tangent cone. 2010 Mathematics Subject Classification : 65K10.
1. I NTRODUCTION The geometric properties of convex closed cones play an important role in theories and algorithms for optimization problems. In particular, the tangent and critical cones are widely used in determining the differential properties of metric projectors and in sensitivity analysis. Moreover, they are necessary for the design of the augmented Lagrangian algorithm. In this paper, we study the geometric properties of a class of cones: the intersection of a closed half-space and a variable box, denoted C := {(y, τ ) ∈ Rn × R : 0 ≤ y ≤ τ e, eT y ≤ κτ },
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where κ > 0 and e denotes the vector of which all elements are 1’s. C appears widely in optimization problems related to the vector k-norm functions and the matrix Ky Fan k-norm functions [1, 2]. For each (y, τ ) ∈ C, it is clear that (y, τ ) satisfies 0 ≤ y ≤ τ e, then the constraint eT y ≤ κτ is not binding if κ ≥ n. In this case, C degenerates into {(y, τ ) ∈ Rn × R : 0 ≤ y ≤ τ e}. In this paper, we restrict 0 < κ < n. There are two motivations of our work. One is the fact that geometric properties are associated with the sensitivity analysis of optimization problems. For nonlinear semidefinite programming problems, Sun researched the relationship among the strong second-order sufficient condition, constraint nondegeneracy, B-subdifferential nonsingularity of the KKT system, and the strong regularity of KKT points [3]. Hence, geometric properties, including the tangent cone, its linear space, the critical cone, and its affine hull, are necessary for sensitivity analysis. In our future work,
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