A generalized second-order derivative in nonsmooth optimization
SIAM Journal on Control and Optimization
Derivatives of spectral functions
Mathematics of Operations Research
Second-order directional derivatives of all eigenvalues of a symmetric matrix
Nonlinear Analysis: Theory, Methods & Applications
Twice Differentiable Spectral Functions
SIAM Journal on Matrix Analysis and Applications
Semismoothness of Spectral Functions
SIAM Journal on Matrix Analysis and Applications
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A spectral function of a symmetric matrix X is a function which depends only on the eigenvalues of X, @l"1 (X) = @l"2(X) == @l"n (X), and may be written as @? (@l"1 (X), @l"2(X), , @l"n (X)) for some symmetric function @?. In this paper, we assume that @? is a C^1^,^1 function and discuss second-order directional derivatives of such a spectral function. We obtain an explicit expression of second-order directional derivative for the spectral function.