Set-Valued and Variational Analysis

, Volume 25, Issue 2, pp 265–296 | Cite as

Variational Analysis of the Ky Fan k-norm

  • Chao DingEmail author


In this paper, we will study some variational properties of the Ky Fan k-norm 𝜃 = ∥⋅∥(k) of matrices, which are closed related to a class of basic nonlinear optimization problems involving the Ky Fan k-norm. In particular, for the basic nonlinear optimization problems, we will introduce the concept of nondegeneracy, strict complementarity and the critical cones associated with the generalized equations. Finally, we present the explicit formulas of the conjugate function of the parabolic second order directional derivative of 𝜃, which will be referred to as the sigma term of the second order optimality conditions. The results obtained in this paper provide the necessary theoretical foundations for future work on sensitivity and stability analysis of the nonlinear optimization problems involving the Ky Fan k-norm.


Ky Fan k-norm Nondegeneracy Critical cone Second order tangent sets 

Mathematics Subject Classification (2010)

65K10 90C25 90C33 


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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institute of Applied Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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