How do find the optimal values for these expressions?

Say I have three constants a, b and c (all > 0) and three variables x, y and z (all >= 0). I want to find values for x, y, z which maximise the lowest of the following:

ax - y - z
-x + by - z
-x -y + cz


How can I find these values?

EDIT: just corrected expressions -- variables and constants were wrong way around

EDIT: x, y and z all must be >=0

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If I understand, you are trying to solve a minimax (or rather, maximin) type problem. Formally, your statement is

$\max \min \{ax-y-z, -x+by-z, -x-y+cz\}$, and you have not listed any other constraints.

A standard trick is to reformulate this problem as a constrained maximization problem. We can do this by introducing an auxiliary variable $\gamma$,

$\max_{\gamma, x,y,z} \gamma$ subject to $\gamma \leq f_i(x,y,z)$ for $i=1,2,3$

where $f_i(x,y,z)$ corresponds to each of the linear functions you stated.

This is nothing but a linear program now, which can be solved using your favorite method.

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Yes that's it -- but I'm afraid I don't now how to solve linear programs -- please could you explain? – user7694 Mar 2 '11 at 13:39
Sorry, but I think "how to solve linear programs" is really beyond the scope of this board. I can only suggest finding a book (Convex Optimization by Boyd is good) or starting with wiki (en.wikipedia.org/wiki/Linear_programming). If you know how to use MATLAB, there are built in solvers (linprog). – 1yen Mar 2 '11 at 13:52
Okay, I'll read up. Thanks for the answer! – user7694 Mar 2 '11 at 13:52
A good book is Dantiz and Thapa's "Linear Programming". – Dilawar Mar 2 '11 at 14:18