Re: Equal constraints for decision values

[Johan Löfberg]

A naive model would be

intvar a

desired = [ zeros(1,34) a a  1.05*a 1.05*a 1.1*a 1.1*a];

sol=optimize([F, sort(I)==desired],f);

Probably very inefficient compared to something smarter

[bin sun]

Dear Johan,

Thank you very much. So I think my code should be:

d=[1*40 matrix];

c=[40*40 matrix]

I=intvar(1,40);  

pos=binvar(1,40);  

intvar a;      %new here

f=-sum(I);

desired = [ zeros(1,34) a a  1.05*a 1.05*a 1.1*a 1.1*a];  %new here

%pos==1, I==0, means empty

F=[implies(pos,I== 0)]; 

F=F+[pos(1:2:39) + pos(2:2:40) >= 1];  % nearby two have one or two empty, empty is 1. So sum is 1 or 2 here.

sum(pos(1:2:39))==17   %type I have: 17 empty so 3 not empty

sum(pos(2:2:40))==17   %type II have: 17 empty so 3 not empty

F=F+[(c*(I.^2)')./d' <= 1];  % important constraint

F=F+[sort(I)==desired];    %new here

sol=optimize(F,f);

I=value(I)   

pos=value(pos)

[bin sun]

So if the new constraint is the 6 I are all equal, I should write something like:

intvar a

desired = [ zeros(1,34) a a  a a a a];

sol=optimize([F, sort(I)==desired],f);

or this will have some simple method to write?

Thank you very much for your time,

Bin

[Johan Löfberg]

Yes, nothing special

[Johan Löfberg]

should be possible to simplify to ismember(I,[0 a])

[bin sun]

Thank you very much. So I think my code should be:

d=[1*40 matrix];

c=[40*40 matrix]

I=intvar(1,40);  

pos=binvar(1,40);  

intvar a;      %new here

f=-sum(I);

%pos==1, I==0, means empty

F=[implies(pos,I== 0)]; 

F=F+[pos(1:2:39) + pos(2:2:40) >= 1];  % nearby two have one or two empty, empty is 1. So sum is 1 or 2 here.

sum(pos(1:2:39))==17   %type I have: 17 empty so 3 not empty

sum(pos(2:2:40))==17   %type II have: 17 empty so 3 not empty

F=F+[(c*(I.^2)')./d' <= 1];  % important constraint

F=F+[ ismember(I,[0 a])];    %new here

sol=optimize(F,f);

I=value(I)   

pos=value(pos)

[Johan Löfberg]

sure.

but you know that you don't have to place things in any particular order. the code I showed you is fully valid

However, a possible alternative is  (might be more efficient)

case1 = binvar(1,40);

case2 = binvar(1,40);

case3 = binvar(1,40);

with constraints

[sum(case1) == 2, implies(case1, I == a), sum(case2) == 2, implies(case2, I == 1.05*a), etc

[bin sun]

Since you said. After I run the code. It's very slow. If I want to accelerate it. Do you have any method to recommend?

[bin sun]

Ok. understand. So basically, in order to improve the efficiency, I need to use 3 decision variables rather than one and reorder them.

[Johan Löfberg]

no

post the data for testing

[Johan Löfberg]

I have no idea what that means. With sort, you will internally introduce 1600 binary variables. Here you introduce 120 variables. The combinatoric structures are completely different though

[bin sun]

d=[349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826,349261.312324434,772899.067110826];

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I=intvar(1,40);  

pos=binvar(1,40); 

intvar Ibase;     %new

case1 = binvar(1,40);  %new

case2 = binvar(1,40);

case3 = binvar(1,40);

f=-sum(I);

F=[implies(pos,I== 0)]; 

F=F+[pos(1:2:39) + pos(2:2:40) >= 1];  % sum could be 1or2; nearby two have one or two empty.

F=F+[pos(1)+pos(3)+pos(5)+pos(7)+pos(9)+pos(11)+pos(13)+pos(15)+pos(17)+pos(19)+pos(21)+pos(23)+pos(25)+pos(27)+pos(29)+pos(31)+pos(33)+pos(35)+pos(37)+pos(39)==17];

F=F+[pos(2)+pos(4)+pos(6)+pos(8)+pos(10)+pos(12)+pos(14)+pos(16)+pos(18)+pos(20)+pos(22)+pos(24)+pos(26)+pos(28)+pos(30)+pos(32)+pos(34)+pos(36)+pos(38)+pos(40)==17];

F=F+[(c*(I.^2)')./d' <= 1]; 

F=F+[sum(case1) == 2, implies(case1, I == Ibase), sum(case2) == 2, implies(case2, I == 1.05*Ibase), sum(case3) == 2, implies(case3, I == 1.1*Ibase)];      %new

sol=optimize(F,f);

I=value(I)   

pos=value(pos)

So case is some new decision variable, not pos, right?

[bin sun]

Dear Johan,

So I rewrite the code like this:

d=[1*40 matrix];

c=[40*40 matrix];

I=intvar(1,40);  

pos=binvar(1,40); 

intvar Ibase;     %new

case1 = binvar(1,40);  %new

case2 = binvar(1,40);

case3 = binvar(1,40);

f=-sum(I);

F=[implies(pos,I== 0)]; 

F=F+[pos(1:2:39) + pos(2:2:40) >= 1];  % sum could be 1or2; nearby two have one or two empty.

F=F+[pos(1)+pos(3)+pos(5)+pos(7)+pos(9)+pos(11)+pos(13)+pos(15)+pos(17)+pos(19)+pos(21)+pos(23)+pos(25)+pos(27)+pos(29)+pos(31)+pos(33)+pos(35)+pos(37)+pos(39)==17];

F=F+[pos(2)+pos(4)+pos(6)+pos(8)+pos(10)+pos(12)+pos(14)+pos(16)+pos(18)+pos(20)+pos(22)+pos(24)+pos(26)+pos(28)+pos(30)+pos(32)+pos(34)+pos(36)+pos(38)+pos(40)==17];

F=F+[(c*(I.^2)')./d' <= 1]; 

F=F+[sum(case1) == 2, implies(case1, I == Ibase), sum(case2) == 2, implies(case2, I == 1.05*Ibase), sum(case3) == 2, implies(case3, I == 1.1*Ibase)];      %new

sol=optimize(F,f);

I=value(I)   

pos=value(pos)

Ibase=value(Ibase)    %new

Is this code looks correct?

When I use the code, it's much faster. But I get the result like: 

I=[0 403 0 0 0 0 0 0 405 0 0 385 0 0 0 0 0 0 354 0 0 0 0 0 0 0 0 0 0 0 0 396 0 0 0 0 0 0 419 0]

Ibase=0;

So it didn't give me the result I need here.

One more thing,  When I  write code like: F=F+[ ismember(I,[0 a])];  MATLAB said:"Both arguments must be constraints". So can you tell me how to write this correctly?

[Johan Löfberg]

The problem is probably the poor numerics as you don't have any bounds on I leading to nasty big-M models

Bounds are easily derived with

[box,L,U] = boundingbox((c*(I.^2)')./d' <= 1,sdpsettings('relax',2,'verbose',0))

and then adding either box, or simply L<= I<=U, and L/1.1<= Ibase <= U/1.1 

You got various other things to play with. For instance, your model implies ismember(Ibase,20:20:U) since 1.05*Ibase isn't an integer otherwise, you also must have case1+case2+case3 == 1-pos

[Johan Löfberg]

and Ibase >= 1 (which means >=20 here just as well) should be added to as those variables shouldn't be 0

[Johan Löfberg]

should be Ibase of course

[bin sun]

So I will have something like this:

d=[1*40 matrix];

c=[40*40 matrix];

I=intvar(1,40);  

pos=binvar(1,40); 

intvar Ibase;     %new

case1 = binvar(1,40);  %new

case2 = binvar(1,40);

case3 = binvar(1,40);

f=-sum(I);

F=[implies(pos,I== 0)]; 

F=F+[pos(1:2:39) + pos(2:2:40) >= 1];  % sum could be 1or2; nearby two have one or two empty.

F=F+[pos(1)+pos(3)+pos(5)+pos(7)+pos(9)+pos(11)+pos(13)+pos(15)+pos(17)+pos(19)+pos(21)+pos(23)+pos(25)+pos(27)+pos(29)+pos(31)+pos(33)+pos(35)+pos(37)+pos(39)==17];

F=F+[pos(2)+pos(4)+pos(6)+pos(8)+pos(10)+pos(12)+pos(14)+pos(16)+pos(18)+pos(20)+pos(22)+pos(24)+pos(26)+pos(28)+pos(30)+pos(32)+pos(34)+pos(36)+pos(38)+pos(40)==17];

F=F+[(c*(I.^2)')./d' <= 1]; 

F=F+[sum(case1) == 2, implies(case1, I == Ibase), sum(case2) == 2, implies(case2, I == 1.05*Ibase), sum(case3) == 2, implies(case3, I == 1.1*Ibase)];      %new

F=F+[1<=Ibase<=400];  %new

F=F+[case1+case2+case3==1-pos];

sol=optimize(F,f);

I=value(I)   

pos=value(pos)

Ibase=value(Ibase)    %new

Is this code looks correct?

I try to add constraint like: F=F+[1<=I<=400]; but MATLAB said "the model is infeasible"

I don't really understand the part of "case1+case2+case3 == 1-pos" could you explain what this mean?

[bin sun]

No matter I use this code or use sort(I), my optimization all stop at 11.1% and stay there like this. Is that possible to use the code to find the result?

[Johan Löfberg]

of course it is infeasible, most of the elements in I are supposed to be zero

and the upper bound on Ibase set to 400, is that based on a-priori knowledge? When I computed the bounding box, it was close to 600

case1+case2+case3 being 1 means that the value is a,1.05a or 1.1*a, i.e. not zero, which you already encode with the pos indicator

[Johan Löfberg]

you tweak the various stopping option in your solver, such as relative gap, absolute gap etc

[Johan Löfberg]

I have no idea if it will good results. It models what you want it to model, but if it does that in a good way is of course impossible to answer

600 came from the boundingbox computations earlier

You should add as many valid constraints you can, that's the whole trick in integer programming.