Local convergence of an augmented Lagrangian method for matrix inequality constrained programming
Optimization Methods & Software
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Real valued functions, $F(X)$, on a symmetric matrix argument are called spectral if $F(U^TXU) = F(X)$ for every orthogonal matrix $U$ and $X \in \mathrm{dom\,} F$. We are interested in a description of the higher order derivatives (when they exist) of $F$ with respect to $X$. Formulae for the gradient and the Hessian of $F$ are given in [A. S. Lewis, Math. Oper. Res., 21 (1996), pp. 576-588] and [A. S. Lewis and H. S. Sendov, SIAM Matrix Anal. Appl., 23 (2001), pp. 368-386]. In this work we present common features of these two formulae that are preserved in the higher order derivatives.