Let $\mathbf c \in \mathbb{R}^n$ where $\mathbf c \neq \mathbf 0$ and $\mathbf A \in \mathbb{R}^{m \times n}$. What is the most efficient way to solve the following optimization problem? Even better, does it have an analytical solution?
\begin{equation} \begin{aligned} \max_{\mathbf x \in \mathbb{R}^n} & &&\mathbf c^\text{T}\mathbf x \\ \text{s.t.} & && \mathbf A \mathbf x \geq \mathbf 0, \\ & && \mathbf x\geq \mathbf 0, \\ & && ||\mathbf x||_2 \leq 1 \end{aligned} \end{equation}
I am interested in the case where $\mathbf x = \mathbf c / ||\mathbf c||$ is not a feasible solution for the problem.
I don't think there is analytical solution.
But you can rewrite it as (Since you limited the norm any way):
$$\begin{aligned} \arg \min_{x} \quad & \frac{1}{2} {\left\| x - c \right\|}_{2}^{2} \\ \text{subject to} \quad & A x \succeq \boldsymbol{0} \\ & x \succeq \boldsymbol{0} \\ & {\left\| x \right\|}_{2} \leq 1 \end{aligned}$$
The it becomes pretty close to your other question - Projection onto a Polyhedral Cone as Minimization of Different Norms with only one more set to project upon.
I took the code I created there and added the projection onto the Euclidean ($ {L}_{2} $ Norm) ball as in Orthogonal Projection onto the $ {L}_{2} $ Unit Ball.
The MATLAB Code which is accessible in my StackExchange Mathematics Q3599003 GitHub Repository.