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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.

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  • $\begingroup$ How big it is? What happens if the solution is indeed what you say isn't feasible? $\endgroup$ – Royi Mar 28 at 22:48
  • $\begingroup$ @Royi, I'm interested in arbitrarily large instances of the problem. But, for concreteness, let's say $n \approx 10^6$ and $m \approx 10^6$. $\endgroup$ – Garrett Mar 29 at 5:00
  • $\begingroup$ If the solution is $\mathbf x = \mathbf c / ||\mathbf c||$, that's great; we're done. But that's not the interesting case. $\endgroup$ – Garrett Mar 29 at 5:02
  • $\begingroup$ Could we assume the norm of $ \boldsymbol{c} $ is 1? Have you looked at my solution? $\endgroup$ – Royi Apr 11 at 23:53
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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.

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