It is well known result of linear optimization theory that, if a finite set $S$ of linear inequalities in real variables $x\_1,x\_2, \ldots ,x\_n$ implies a linear inequality $i$ in $x\_1,x\_2, \ldots ,x\_n$, then $i$ can be written as a positive linear combination of the inequalities in $S$.


I wonder if an analogous property holds for moduluses of complex variables : let $A=(a\_{ij}) (1 \leq i \leq p, 1 \leq j \leq n)$ be a matrix in ${\cal M}\_{p,n}({\mathbb C})$ and $B=(b\_1,b\_2, \ldots b\_p)^{T}$ be a column matrix in ${\cal M}\_{p,1}({\mathbb R}^{+})$. Then we may denote by $|AZ| \leq B$ (where $Z=(z\_1,z\_2, \ldots z\_n)^{T}$ is a column matrix in ${\cal M}\_{n,1}({\mathbb C})$ ) the finite set of modulus constraints $\big| \sum\_{j=1}^{n}a\_{ij}z\_j \big| \leq b\_i$ for $1\leq i\leq p$. Suppose this set of constraints implies another modulus constraint (\*), say $\big| \sum\_{j=1}^{n}c\_{j}z\_j \big| \leq d$, with $c\_j \in {\mathbb C}$ and $d \geq 0$.


The question, then, is : can (\*) be deduced linearly from $S$ ? In other words, are there scalars ${\lambda}\_1, {\lambda}\_2, \ldots ,{\lambda}\_p$ in $\mathbb C$ with


$$
\sum\_{i=1}^{p}\lambda\_ia\_{ij}=c\_j \ ({\rm for} \ 1 \leq j \leq n), \ \
\sum\_{i=1}^{p}|\lambda\_i|b\_i \leq d.
$$



Complex version of Farkas' lemma
I think, yes. Without loss of generality, all $b\_i$'s and $d$ are equal to 1. Assume that the vector $c=(c\_1,\dots,c\_n)$ does not lie in a convex hull of vectors $(wa\_{i,1},\dots,wa\_{i,n})$ for all $i=1,\dots,n$ and all $|w|=1$. Then there is a linear real functional, which separates $c$ from this convex hull. It may be written as $F(c)= \Re(u\_1c\_1+\dots+u\_nc\_n)$. But if we maximize $F(wa)$ for fixed vector $a$ and various $w$ on the unit circle, we see that the maximum equals $|u\_1a\_1+\dots +u\_na\_n|$. So, this hypothetic separation is exactly the contrary to our assumption.


