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How can I minimize or maximize a function (even numerically) with logical arguments, taking only 0s and 1s, also with constraints? E.g.

NMinimize[{x1 * 24 + x2*51 + x1*x2 *36, 12 x1+36 x2<44}, {x1,x2}]

The example of course is stupid one, but question as a matter of principle, when you have many arguments.

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  • $\begingroup$ Min[Pick[Outer[Function[{x1, x2}, x1*24 + x2*51 + x1*x2*36], {0, 1}, {0, 1}], Outer[Function[{x1, x2}, 12 x1 + 36 x2 < 44], {0, 1}, {0, 1}]]]? $\endgroup$ – J. M. will be back soon Nov 3 '15 at 8:26
  • $\begingroup$ I need something general, true for many arguments. $\endgroup$ – Al Guy Nov 3 '15 at 8:27
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    $\begingroup$ Min[Pick[Function[{x1, x2}, x1*24 + x2*51 + x1*x2*36] @@@ Tuples[{0, 1}, 2], Function[{x1, x2}, 12 x1 + 36 x2 < 44] @@@ Tuples[{0, 1}, 2]]]. Adjust the second argument of Tuples[] as seen fit. $\endgroup$ – J. M. will be back soon Nov 3 '15 at 8:30
  • $\begingroup$ Great! I was just thinking there should be some special ways for categorial optimization. $\endgroup$ – Al Guy Nov 3 '15 at 8:40
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    $\begingroup$ How about the following? NMinimize[{x1*24 + x2*51 + x1*x2*36, 12 x1 + 36 x2 < 44 && And @@ (0 <= # <= 1 && # \[Element] Integers & /@ {x1, x2})}, {x1, x2}] $\endgroup$ – kirma Nov 3 '15 at 15:39
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You could use $v^2 = v$ as a constraint. For your example:

NMinimize[
    {
    x1*24 + x2*51 + x1*x2*36,
    12 x1+36 x2<44 && x1^2==x1 && x2^2==x2},
    {x1,x2},
    Integers
]

{0., {x1 -> 0, x2 -> 0}}

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  • $\begingroup$ Alternatively, add constraint 0<=x1<=1 and similar for x2. And keep the Integers domain specification. Not sure which of these will perform better in general so might want to experiment. $\endgroup$ – Daniel Lichtblau Dec 13 '17 at 23:31

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