How to add constraint with condtions using fmincon?

I have am optimization problem with two variables x and y where x and y are vectors.

I would like to add the foollowing constraint to fmincon :

if  a<b then x<y

where a and b are known values.

I would like also to add in general x<y for every value of both vectors.

Many thanks!

Answer 1

[xi]

Your constraint could be written as:

x-y < 1/(a<b)-1

when a<b, a<b=1, so, 1/(a<b)-1=0

when a>=b, a<b=0, so, 1/(a<b)-1=Inf, it is equivalent to not having constraint.

[wiem abd]

Thanks for your answer! However I don't understand what (a<b)-1 means. Can you explain it to me?

[xi]

Try these in the command line, and you will find out the answer

a=1;b=2;

a<b

1/(a<b)-1

then switch a and b, try again.

a<b returns the logical true(1) or false(0)

[wiem abd]

Thank you! What if I want to add constraint where x(i)<y(i) for in {1,length(x)}

[xi]

Assuming your x and y array length is n, you have 2n variables, not 2. So

X= (x1,x2,...xn,y1,y2...yn)

The constraint should be written in the vectorized format of AX-b<0;

for example, x1<y1 and x2<y2,... should be written as

A= [ 1,0,0...-1,0,0,...;

0,1,0...0,-1,0,...;

...]

b= [0,0,...]

I don't quite understand what in {1,length(x)} means. You can either use the trick I described and write A in an (n by 2n) matrix, or find out how many total contraints (=m) you have, and write A in an (m by 2n) matrix, where m<n.

[wiem abd]

Thank you ! That was exactly what I needed.

Now let us consider that I have a vector L of lenght N where the known values are the elements of L.

I want to check for the vector X of lenght N that

for i=1:N

    for j=1:N

    if L(i)<L(j) then X(i)<=X(j) or X(i)-X(j)<1/(L(i)<L(j))-1
        
    end
end

How to translate this into Matrix because I need to insert it as a constraint in fmincon

Answer 2

[xi]

Ask yourself 3 question: How many vairable? how many constraints? and then how to write A and b.

Now you are saying L is a known vector, then, your vairables are just X of length N. you can write your constraint in this way:

-----------------------------

A=zeros(N^2, N);

b=ones(N^2,1);

count=0;

for i=1:N

for j=1:N

count=count+1;

A(count,i)=1;

A(count,j)=-1;

b(count)= 1/(L(i)<L(j))-1;

end

end

---------------------------------

or using N(N-1)/2 constraints instead of N^2

for i=1:N-1

for j=i:N

................

end

end

---------------------------------------

A better way is to sort L first, and get the ordering index using [~,index] = sort(L)

So you define your new variable X'=X(index); and solve X' instead.

then, you only need to write N-1 constraints:

X'(1)<X'(2), X'(2)<X'(3), ... X'(N-1)<X'(N)

b is simply b=zeros(N-1,1); You can figure out A. This should be much faster.

[wiem abd]

Thank you very much !

1. Yes I thought about sorting L, but I cannot define X'=X(index); and solve X' instead because the rest of the constraints depend on X.
2. Regarding "using N(N-1)/2 constraints instead of N^2", I think it is only possible when L is sorted. Isn't?
3. Do you mean to use the loop body of N^2 constraints as follows :

for i=1:N-1
   for j=i:N
         count=count+1;    
      A(count,i)=1;
      A(count,j)=-1;  
      b(count)= 1/(L(i)<L(j))-1;     
   end
end

[xi]

forget about solution 2，I was wrong， i,j are not switchable. Solution 3 is really what you should go for:

Just One loop over i=1:N-1 A(count,index(i))=1; A(count,index(i+1))=-1;

[wiem abd]

Do you mean I can use it without replacing X with X(index)?

Is the following code correct in this case ?

 [~,index] = sort(L)
 
 for i=1:N-1 
     A(count,index(i))=1; 
     A(count,index(i+1))=-1;
 end
 
 b=zeros(N-1,1); 

[wiem abd]

This is the obtained result for N=10. I thing I am missing something

index =

     5
     3
     2
    10
     1
     6
     8
     4
     7
     9

>> A

A =

     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0
     1     1     1     1     1     1     1     1    -1     1

>> size(A)

ans =

   100    10

[xi]

no longer need count, just A(i, index(i)), or you need count++

[wiem abd]

Thank you very much.

It is working correctly now.

Answer 3

[wiem abd]

Now let us consider that my variable X is of length 2*N and my known values are in a vector L of lenght N.

I want that :

for i=1:N
    for j=1:N
    if L(i)<L(j) then X(i)<=X(j) and X(N+i)<X(N+j)
        
    end
end

How can I find A such that A*X<zeros(N-1,1).

[xi]

Now you have 2N variables, and 2*(N-1) constraints, only need small modifications of the above code

A=zeros(2*(N-1), 2*N);

[~,index] = sort(L)

for i=1:N-1 % write two constraints for each i

A(i,index(i))=1;

A(i,index(i+1))=-1;

% ******* just add two lines here, I leave it to you to figure out what the index of A should be.

end

b=zeros(N-1,1);

[wiem abd]

I think these should be the lines to add :

 A(i+N,N+index(i))=1;
     A(i+N,N+index(i+1))=-1;

b size should also be modified as follows :

      b=zeros(2*N-1,1); 

It is working and giving correct results .

Many thanks !

Answer 4

[wiem abd]

Is there a way to formulate this anlytically using equations ?