MaplePrimes - answers and comments on Question, Optimization problem

Solution

Kitonum — Thu, 03 Jan 2013 22:01:17 Z

Your problem is easily solved by exhaustive search. Denoted by a, b, c, d, e, f, g, h vertices of the cube:

Code:

L:=combinat[permute]({seq(i^2, i=1..8)}):

n:=infinity: S:={}:

for i in L do

a, b, c, d, e, f, g, h := i[1], i[2], i[3], i[4], i[5], i[6], i[7], i[8]:

m := a*b+b*c+c*d+a*d+a*e+b*f+c*g+d*h+e*f+f*g+g*h+e*h: S:=S union {m}:

if m<n then n:=m:

M:=[m, ["a"=a, "b"=b,"c"= c, "d"=d, "e"=e, "f"=f, "g"=g, "h"=h]]: fi:

od:

S: # The set of all possible values of this sum

nops(S);

max(S);

min(S);

622

9420

3586

[3586, ["a" = 1, "b" = 36, "c" = 9, "d" = 64, "e" = 49, "f" = 16, "g" = 25, "h" = 4]]

Solution with DirectSearch package

Kitonum — Fri, 04 Jan 2013 18:08:44 Z

On the solution of discrete optimization problems, see ?DirectSearch[Search].

Solution of the problem:

The example shows that the solution of such problems with DirectSearch package is not effectively.

integer.mw

Vote up

Markiyan Hirnyk — Thu, 03 Jan 2013 22:08:44 Z

It is sure unexpected. I tried to formulate and to solve an integer programming problem.

PS. I have strong doubts that the four-dimensional version of that problem can be solved
by item-by-item examination: 16! = 20922789888000.

Re

Kitonum — Thu, 03 Jan 2013 22:55:03 Z

@Markiyan Hirnyk

If you know of another way to solve this problem (for example, as integer programming problem), please show your code.

Integer programming problem

Markiyan Hirnyk — Thu, 03 Jan 2013 23:26:38 Z

@Kitonum I am working with

minimize f = x[1]*x[2]+x[1]*x[5]+x[1]*x[4]+x[3]*x[2]+x[3]*x[4]+x[3]*x[7]+x[8]*x[5]+x[8]*x[7]+
x[8]*x[4]+x[6]*x[5]+x[6]*x[7]+x[6]*x[2]

under the constraints

{seq(product(j^2-x[k], j = 1 .. 8) = 0, k = 1 .. 8),
sum(x[j], j = 1 .. 8) = add(j^2, j = 1 .. 8)}

assuming posint.

My attempt

Markiyan Hirnyk — Thu, 03 Jan 2013 23:41:11 Z

>DirectSearch:-GlobalOptima(x[1]*x[2]+x[1]*x[5]+x[1]*x[4]+x[3]*x[2]+x[3]*x[4]+x[3]*x[7]+x[8]*x[5]+x[8]*x[7]+x[8]*x[4]+x[6]*x[5]+x[6]*x[7]+x[6]*x[2], {seq(product(j^2-x[k], j = 1 .. 8) = 0, k = 1 .. 8), sum(x[j], j = 1 .. 8) = add(j^2, j = 1 .. 8)}, assume = posint);

[HFloat(7173.0), [x[1] = 25, x[2] = 4, x[3] = 9, x[4] = 64,

x[5] = 4, x[6] = 36, x[7] = 25, x[8] = 36], 2090]

If Sergey Moiseev finds time to look at that problem, it would be very kind of him.