TITLE: How do I solve $\min_x \max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)$ for $\lVert x \rVert_2 = 1$.
QUESTION [15 upvotes]: Let $f(x) = \max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)$.
where $x, c_1, c_2, \dots, c_k \in \mathbb R^n$.
What fast iterative methods are available for finding the (approximate) min of $f$ with the constraint $\lVert x \rVert_2 = 1$?
Notes:

$f$ is convex and and non negative, that is, $c_i$ positively span $\mathbb R^n$
For my use case $k \gg n$ and (rougly) $3 \le n \le 50$ and $100 \le k \le 10000$.
I tried projected subgradients (projected to the sphere) but it can be slow to converge and improving the initial guess doesn't seem to accelerate the method. From the literature, this seems to be equivalent to Riemannian manifold subgradient methods which use the exponential map. I haven't tried bundle methods yet but I am investigating them now. They seem like they might be too slow.

I've also tried approximating $f$ with a smooth maximum LogSumExp and doing gradient descent with projection. I also tried unconstrained gradient descent with a quadratic penalty. The approximation is too inaccurate and the evaluation of exp is too slow for my use case. I'm not interested in SQP because I suspect whatever method used to smooth (e.g. LogSumExp, hyperbolic) will be too costly/inaccurate to evaluate.
Edit:
I believe this problem is equivalent to a linear programming problem with quadratic equality constraints. I haven't been able to to find much literature on this type of problem. Perhaps I can use a SDP method but I'm not sure.

REPLY [3 votes]: We need to solve the following optimization problem (P$0$):
\begin{align}
(\mathrm{P0})\qquad &\min_{x} \max\{c_1^\mathsf{T}x, \cdots, c_k^\mathsf{T}x\}\\
&\mathrm{s.t.}\quad \|x\|_2 = 1.
\end{align}
Assume that $\{c_1, c_2, \cdots, c_k\}$ satisfies
$\max\{c_1^\mathsf{T}x, \cdots, c_k^\mathsf{T}x\} > 0$ on $\mathbb{R}^n\backslash \{0\}$.
(P$0$) is equivalent to $\min_{x} \frac{\max\{c_1^\mathsf{T}x, \cdots, c_k^\mathsf{T}x\}}{\|x\|_2}$
and $\max_{x} \frac{\|x\|_2}{\max\{c_1^\mathsf{T}x, \cdots, c_k^\mathsf{T}x\}}$ and
\begin{align}
&\max_{x} \|x\|_2^2\\
&\mathrm{s.t.}\quad \max\{c_1^\mathsf{T}x, \cdots, c_k^\mathsf{T}x\} = 1
\end{align}
and
\begin{align}
(\mathrm{P1})\qquad &\max_{x} \|x\|_2^2\\
&\mathrm{s.t.}\quad c_i^\mathsf{T}x \le 1, \ i=1, 2, \cdots, k.
\end{align}
(P$1$) is a Quadratic Programming (QP) problem (https://en.wikipedia.org/wiki/Quadratic_programming).
Some online and software Resources for QP are given in [1]. Some quadratic programming codes are given in [2].
References
[1] https://neos-guide.org/content/quadratic-programming
[2] http://www.numerical.rl.ac.uk/people/nimg/qp/qp.html