Suppose that $A(x) \in \mathbb{R}^{m \times n}$ is a nonlinear matrix function of $x \in \mathbb{R}^d$. We may assume that $A(x)$ is continuously differentiable. Are there any good ways to estimate $\mathrm{argmin}_{x} \|A(x)\|_2$? Also assume that due to some growth properties of the function $A(x)$, this minimizer can only be achieved within a known compact set. There are well known methods when $A(x)$ is linear with respect to $x$, but I don't have that linearity here.I have tried to implement a few quasi-Newton methods with varying success, and I've also tried MATLAB's fminunc command with reasonable success (stabilizing around local minima is a problem). Unfortunately I'm relatively unfamiliar with the literature on these things (and the literature is quite vast). Are there any general approaches anybody could suggest?
Nonlinear optimization for minimizing matrix norm
optimization
This seems to be exactly the problem addressed in the following paper. Just recast the problem of minimizing $||A||_2$ into a problem of minimizing the maximum eigenvalue of $A^T A$.Overton, Michael L., and Robert S. Womersley. Second derivatives for optimizing eigenvalues of symmetric matrices. SIAM Journal on Matrix Analysis and Applications 16.3 (1995): 697-718.  http://ftp.cs.nyu.edu/cs/faculty/overton/papers/pdffiles/eighess.pdf