# minimization problem with integer variables and constraints

My question is this:

Find l,m,k (l,m,k are integers $>$0 and l is odd) which minimize:

$M(l,m,k)=m^2(1-\gamma)+\frac{l^2}{4}\gamma-k^2$

(Where $0<\gamma<1$ and $\gamma$ is a known rational number.)

Under the constraint of:

$(m-\frac{l}{2})^2<\frac{M(l,m,k)}{\gamma(1-\gamma)}<(m+\frac{l}{2})^2$

• A code solving this problem is also welcomed. – dr.bian May 9 '17 at 4:48
• l,m,k are independent of each other, so a way to do this is choosing m,l as small as possible while choosing k as large as possible. – dr.bian May 9 '17 at 4:50

Your constraint does'nt match together with the other conditions.

That means, there is no solution

 M[l_, m_, k_] = m^2 (1 - Gamma) + l^2/4 Gamma - k^2

constraint = (m - l/2)^2 < M[l, m, k]/(Gamma*(1 - Gamma)) < (m + l/2)^2

Reduce[
constraint && 0 < Gamma < 1 && l > 0 && m > 0 && k > 0 &&
m \[Element] Integers && l \[Element] Integers, k, Integers]

(*    False    *)

• Thanks for your answer. This is a mathematica code and it is really helpful, but the last two element in reduce, I think they should be written as && k[Element] Integers – dr.bian May 9 '17 at 16:15
• Reduce[constraint && 0 < Gamma < 1 && l > 0 && m > 0 && k > 0 && m [Element] Integers && l [Element] Integers && k [Element] Integers]? – dr.bian May 9 '17 at 16:16