Find maximum of the function

Question

I have the following target function

$$ f(m,q)=\sum^{N}_{i=1}|m_i-q_i| $$, where $$m,q\in R^N$$ and $$\sum^{N}_{i=1}m_i=1, \forall m_i>0$$ $$\sum^{N}_{i=1}q_i=1, \forall q_i>0$$

I would like to find such vectors $$q,m$$ that deliver maximum of the target function.

My initial guess - any vectors $$m,q$$ will be the solution of the defined above problem if $$\sum^{N}_{i}m_i=1, \sum^{N}_{i}q_i=0$$

The logic is that the set described by the boundaries is compact and the target function is monotonic. Since the maximum will be delivered on the boundaries of the set.

wonko · Answer 1 · 2014-07-16T18:06:27.590

0

You can see this easily by arguing that for any given $m$ with $m_i \geq 0.5$, the maximum is achieved by setting $q$ to zero. This is just restating your logic though and equally not rigorous :)

edited Jul 16 '14 at 18:06

answered Jul 16 '14 at 14:25

wonko

485

Do you meen that the target function doesn't have any maximum? – Taras Murzenkov Jul 16 '14 at 17:21
No. Your answer was right. I was just presenting an alternative explanation, which as it turns out was incomplete so I edited it. – wonko Jul 16 '14 at 18:07

Find maximum of the function

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