Re: Maximize with Integer constraints

• To: mathgroup at smc.vnet.net
• Subject: [mg78247] Re: [mg78227] Maximize with Integer constraints
• Date: Wed, 27 Jun 2007 05:17:25 -0400 (EDT)
• References: <200706260833.EAA05703@smc.vnet.net> <49779893-9F18-4F8C-B0D8-2EB6F23313A7@mimuw.edu.pl>

```Andrzej Kozlowski wrote:
>
> On 26 Jun 2007, at 17:33, sdw wrote:
>
>>
>> given entry #1:
>>
>>
>> Maximize[
>> {x1 + x2 + x3 + x4,
>> Element[x1 | x2 | x3 | x4 , Integers] &&
>> 0 <= x1 <= 5.6 &&
>> 0 <= x2 <= 8.6 &&
>> 0 <= x3 <= 9.7 &&
>> 4.0 <= x4 <= 22.4 }, {x1, x2, x3, x4}]
>>
>> {4., {x1 -> 0, x2 -> 0, x3 -> 0, x4 -> 4}}
>>
>> note -  zeros for answers
>> entry #2:
>> Maximize[
>> {x1 + x2 + x3 + x4,
>> Element[x1 | x2 | x3 | x4 , Integers] &&
>> 0 <= x1 <= 5.6 &&
>> 0 <= x2 <= 8.6 &&
>> 0 <= x3 <= 9.7 &&
>> 4.1 <= x4 <= 22.4 }, {x1, x2, x3, x4}]
>>
>> {44., {x1 -> 5, x2 -> 8, x3 -> 9, x4 -> 22}}
>>
>> only difference is 4.1 vs. 4 in constraints
>> any ideas what is going on?
>>
>> thanks,
>>
>> sdw
>>
>>
>
> This is probably a bug, but in any case, since Maximize uses exact
> algebraic methods it is not a good idea to have approximate numbers in
>
> Maximize[{x1 + x2 + x3 + x4, Element[x1 | x2 | x3 | x4, Integers] &&
>        Rationalize[0 <= x1 <= 5.6 && 0 <= x2 <= 8.6 && 0 <= x3 <= 9.7 &&
>            4. <= x4 <= 22.4]}, {x1, x2, x3, x4}]
>  {44, {x1 -> 5, x2 -> 8, x3 -> 9, x4 -> 22}}
>
> Andrzej Kozlowski
>

When inexact numbers are present in the input, Maximize simply passes
the problem to NMaximize. NMaximize uses numeric methods and is not
guaranteed to find the global maximum (though in this example it
probably should do better). To use exact optimization methods you need
to make sure that the input given to Maximize is exact.

As Andrzej Kozlowski has shown, inexact expressions can be converted to
exact ones using Rationalize. An issue worth remembering here is that
one-argument Rationalize converts only inexact numbers that are "close"
to rationals.

In[1]:= Rationalize[{0.5, 3.1415927}]

1
Out[1]= {-, 3.14159}
2

Rationalize[expr, 0] converts all inexact numbers in expr.

In[2]:= Rationalize[{0.5, 3.1415927}, 0]

1  31415927
Out[2]= {-, --------}
2  10000000

Best Regards,