Mixed Integer Conic Program and semi-definiteness

Problems with modeling
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AMIIT
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Mixed Integer Conic Program and semi-definiteness

Post by AMIIT » 2 months ago

Hey all.
I had previously posted a problem of mine, to which I was advised to re-formulate the problem.
Link to previous problem viewtopic.php?p=28580#p28580

I am trying to formulate the problem given in the screenshot and code, below.
image.png
Pls ignore all the display commands and the disorderliness. I am quite new to GAMS and I am still learning a lot.

Code: Select all

Set
    i /1*8/
    j /1*3/
    g /1,2/;
            
Alias (i,k)
    (i,k1)
    (i,k2)
    (i,k3)
    (i,ip);
Alias  (j,k4)
    (j,k5)
    (j,k6)
    (j,w)
    (j,u);
    
Variables 
          L
          Z1
          cost final cost;
          
Binary variable
          F(i,i) each variable;
cost.l = 350;
F.l(i,i) = 1;
display F.l;

Parameter
IM(i,i) identity matrix;

IM(i,i) = 1;
display IM;


Parameter
C(i, j) C matrix/

1.1 1, 1.2 0, 1.3 0
2.1 1, 2.2 1, 2.3 -1
3.1 1, 3.2 1, 3.3 -1
4.1 1, 4.2 1, 4.3 -1
5.1 0, 5.2 1, 5.3 0
6.1 1, 6.2 0, 6.3 -1
7.1 0, 7.2 1, 7.3 -1
8.1 0, 8.2 0, 8.3 1
/;

display C;

parameter Ct(j,i);

Ct(j,i) = C(i,j);

display Ct;

Parameter Q(i,i);

loop(k1, Q(k1,ip) = sum((i), F.l(k1,i)*IM(i,ip)));
display Q;

Parameter A(w,u) ;

loop(k,  A(w,u) = sum((k2,k3),Ct(w,k3)*Q(k3,k2)*C(k2,u)));

display A;

Parameter invA(j,j);

execute_unload 'a.gdx', j, A;
execute '=invert.exe a.gdx j A b.gdx invA';
execute_load 'b.gdx', invA;

display invA;

parameter AA(i,i);

loop(k6, AA(i,i) = sum((k4,k5), C(i,k5)*invA(k5,k4)*Ct(k4,i)));

display AA;

scalar t trace;
t = sum(i,AA(i,i))

display t;

Parameter Y(i,i) Random variable;

Y(i,i) = 0;

Parameter P(i,i) cost per sensor;

P(i,i) = 100;

Parameter TR;

TR = sum(i, Y(i,i));

equation
    obj
    con1
    con2;
    
obj.. L =e= t/2;
con1.. TR =L= t;
con2.. cost =G= sum((i,ip), F(i,ip)*P(ip,i));

model m /all/;
option qcp = mosek;
m.dictfile = 1;
m.optfile = 1;

$echo SDPSOLUFILE sdpsol.gdx > mosek.opt

Solve m minimizing L using QCP;
display L.l;
Let me explain how the problem works. So basically I need to get values of F11 - F88, which gives me the minimum cost and trace( L value). Also I have given the constraint in such a way that only at most 3 of the F values will be 1 at a time. What the program needs to do is check for all combinations of F11-F88 to get the optimal combination of values. Also the F variable should ideally be a binary variable, as it should only take 0 or 1 as its values.

As you can see from the code, C matrix is defined as such. The F matrix has its diagonal elements as 8 different variables(F11-F88), whose values I've for the moment given as 1. It should actually have different values corresponding to the minimizing equation.
Y is an intermediate variable not important.

My first question is, how do I code the semi-definiteness part? These are the constraints that involve the Y variable. Second, is there any way to optimize this code better?

Thank you in advance.
Attachments
image.png

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