[Couenne] Problem Reformulation (P) and (P')

[Pietro Belotti]

Dear Sismail,

the second problem has many degrees of freedom. A prototype is the  
following problem:

min (x-y)^2

that admits infinitely many solutions (x,y) = (h,h) depending on the  
initial point given to the solver. The NLP solver used by Couenne,  
which is Ipopt, finds a point that is different depending on the  
formulation of the problem.

You may try to solve the first problem and use its optimal solution as  
a starting point for the second (in AMPL, by using the "default"  
keyword). In that case, the two optimal solutions should be very close.

Cheers,
Pietro

_________________________________________
Pietro Belotti, Lehigh University
Dept. of Industrial & Systems Engineering
200 W Packer Ave, Bethlehem PA 18015.
phone: 610-758-3865   fax: 610-758-4886
email: belotti at lehigh.edu
web:   http://www.lehigh.edu/~pib208


On 07/09/2009, ksismail <ksismail1 at gmail.com> wrote:

> Dear Pietro,
>
>       The response is adequate. Just one important thing. Why the variables
> values are so different ? (See the attached file).
>
>       in other words, what to do for getting variables values differences
> more closest ? at least less than 1e-5.
>
> Regards,
>
> Sismail
>
>
> 2009/6/30 Pietro Belotti <belotti at lehigh.edu>
> Dear Sismail,
>
> I'm attaching below the output of Couenne on both instances. The optimal
> solution is zero, and the two "different" numbers obtained are both very
> close to zero -- they are indeed returned by Ipopt, the NLP solver used by
> Couenne. In this case I wouldn't say the two solutions are different.
>
> Also, testb.mod is reformulated by Couenne so that the first variables to
> appear in the AMPL file, testb.mod, i.e., x[6], x[7], and x[8], are indeed
> the first variables in the model file loaded by Couenne. This is again a
> slight difference in the model, but only related to a re-numbering of the
> variables.
>
> Hope this helps.
>
> Best,
> Pietro
>
>
> [pbelotti ~] couenne testa
>
> objectives:
> min ((-0.1+(x_0+x_1+x_2+(2*x_3)+(
>>
>> 2*x_4)+(2*x_5)))^2)
>> constraints:
>> variables:
>> x_0 [ 0 , 1 ]
>> x_1 [ 0 , 1 ]
>> x_2 [ 0 , 1 ]
>> x_3 [ 0 , 1 ]
>> x_4 [ 0 , 1 ]
>> x_5 [ 0 , 1 ]
>> end
>> Problem size before reformulation: 6 variables (0 integer), 0 constraints.
>> Problem size after  reformulation: 8 variables (0 integer), 0 constraints.
>>
>> NLP0012I
>>              Num      Status      Obj             It       time
>> NLP0013I     1        OPT         7.24915497916231e-16    8        0.008001
>> Cbc0012I Integer solution of 7.24915e-16 found by Init Rounding NLP after 0
>> iterations and 0 nodes (0.00 seconds)
>> NLP0013I     2        OPT         1.230414883748645e-20   3        0.004
>> Cbc0001I Search completed - best objective 7.24915497916231e-16, took 0
>> iterations and 0 nodes (0.00 seconds)
>> Cbc0035I Maximum depth 0, 0 variables fixed on reduced cost
>>
>> couenne Optimal
>>
>>        "Finished"
>>
>>
>> [pbelotti ~] couenne testb
>>
>> objectives:
>> min ((-0.1+(x_0+x_1+x_2))^2)
>> constraints:
>> ( +1*x_8 +1*x_7 +1*x_3 -1*x_0) = 0
>> ( +1*x_8 +1*x_6 +1*x_4 -1*x_1) = 0
>> ( +1*x_7 +1*x_6 +1*x_5 -1*x_2) = 0
>> variables:
>> x_0 [ 0 , 3 ]
>> x_1 [ 0 , 3 ]
>> x_2 [ 0 , 3 ]
>> x_3 [ 0 , 1 ]
>> x_4 [ 0 , 1 ]
>> x_5 [ 0 , 1 ]
>> x_6 [ 0 , 1 ]
>> x_7 [ 0 , 1 ]
>> x_8 [ 0 , 1 ]
>> end
>> Problem size before reformulation: 9 variables (0 integer), 3 constraints.
>> Problem size after  reformulation: 11 variables (0 integer), 0 constraints.
>>
>>
>> ******************************************************************************
>> This program contains Ipopt, a library for large-scale nonlinear
>> optimization.
>>  Ipopt is released as open source code under the Common Public License
>> (CPL).
>>         For more information visit http://projects.coin-or.org/Ipopt
>>
>> ******************************************************************************
>>
>> NLP0012I
>>              Num      Status      Obj             It       time
>> NLP0013I     1        OPT         8.626481252066872e-18   4        0.004
>> Cbc0012I Integer solution of 8.62648e-18 found by Init Rounding NLP after 0
>> iterations and 0 nodes (0.00 seconds)
>> NLP0013I     2        OPT         2.593450360578931e-25   4        0.004
>> Cbc0001I Search completed - best objective 8.626481252066872e-18, took 0
>> iterations and 0 nodes (0.00 seconds)
>> Cbc0035I Maximum depth 0, 0 variables fixed on reduced cost
>>
>> couenne Optimal
>>
>>        "Finished"
>>
>



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