## A linear program related question

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Dear all, recently, I encountered the following problem. It is closely related to the order of growth for UMD constants of all $n$-dimensional Banach lattice.

Let $\alpha^k \in (\alpha_1^k, \alpha_2^k, \alpha_3^k, \alpha_4^k) \in [0,1]^4$ for $k = 1, 2, 3, 4$. For $x = (x_1, x_2, x_3, x_4) \in [0,1]^4$, define $$\| x\|_k = \sum_i \alpha_i^k x_i.$$

Then define $$\| x\| = \max(\| x \|_1, \| x \|_2, \|x \|_3, \| x \|_4).$$

Define $$\Large F(x,y, \alpha, \beta, \gamma, \delta) := \frac{\|(\frac{x_1+ y_1}{2}, \frac{x_2+y_2}{2}, x_3, x_4)\| +\|(\frac{x_1+ y_1}{2}, \frac{x_2+y_2}{2}, y_3, y_4)\| }{2 \max (\|(x_1, x_2, x_3, x_4)\|, \|(y_1, y_2, y_3, y_4)\|)}$$

I want to solve the following: find $(x_0, y_0, \alpha_0, \beta_0, \gamma_0, \delta_0)$, $$F(x_0, y_0,\alpha_0, \beta_0, \gamma_0, \delta_0) = \Large \max F(x,y, \alpha, \beta, \gamma, \delta)$$ $$\Large \textbf{s.t.} \alpha, \beta, \gamma, \delta\in [0,1]^4, \quad x, y \in [0, 1]^4, x \ne 0.$$

This is equivalent to $$\max\frac{\|(\frac{x_1+ y_1}{2}, \frac{x_2+y_2}{2}, x_3, x_4)\| +\|(\frac{x_1+ y_1}{2}, \frac{x_2+y_2}{2}, y_3, y_4)\| }{2}$$ $$\Large \textbf{s.t.} \alpha, \beta, \gamma, \delta\in [0,1]^4, \quad x, y \in [0, 1]^4, x \ne 0$$ $$\Large\|(x_1, x_2, x_3, x_4)\| \le 1$$ $$\Large\|(y_1, y_2, y_3, y_4)\| \le 1$$

Probably, this problem is classical in convex program, and maybe we even have software for computing this kind of problem. Are there anyone who can do this by computer (or maybe even by hand)?

$\textbf{Remark}:$ I have tried some example by hand, and I know that $\max F(x,y) \ge \frac{3}{2}$. Theoretically, I also know that $\max F(x,y) \le 2$. The most interesting question is: Do we have $$\max F(x,y) = 2 ?$$ If it is, I can get an optimal estimate of the supremum of (say) $\text{UMD}_2$ constant for all $n$-dimensional Banach lattice.

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Is there some reason that this is convex? – Igor Rivin Aug 12 at 22:09
Firstly, I made a mistake in stating the problem. The reason this is convex is probably clear now. – Yanqi QIU Aug 13 at 6:35
Seems like a quadratic program in less than 30 variables. I would've suggested an SDP relaxation, but you might as well just use a quadratic program solver, of which there are many, check your matlab distribution.  – Sasho Nikolov Aug 13 at 9:05
Thank you for the suggestion. The reason that asked this question is because I have never learned matlab or anything on computer program...   – Yanqi QIU Aug 13 at 12:24
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