[Couenne] equivalence of problem (P) and (P')

[Pietro Belotti]

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"



On 06/30/2009, ksismail <ksismail1 at gmail.com> wrote:

> Thank you, Pietro, for responding.
>
> Please find enclosed 2 files .mod and their equivalent .nl for a very simple
> example for problem(P) and problem(P'). I still didn't find same objective
> values and variables values for the two equivalent problems. Something is
> missing or my Couenne installation is not good.
>
> Regards,
> Sismail
>
>
> 2009/6/19 Pietro Belotti <belotti at lehigh.edu>
> Dear Sismail,
>
> these two problems look equivalent to me, the only difference is that P'
> explicitly introduces auxiliary variables that would be created by Couenne.
>
> I'm not sure of how the g() functions look like, but this is in general a
> nonconvex problem. If you are interested in understanding why the solutions
> you get are different (which is the problem you had in the beginning, if I
> remember correctly), then a more precise formulation is needed. Do you have
> an AMPL model, or a .nl file produced by AMPL?
>
> Hope this helps.
>
> 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 <http://www.lehigh.edu/%7Epib208>
>



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