adding an if condition in yalmip code

[nadiauir]

Hi my name is nadia, i'm PHD student. and i need help about an optimization problem that i wrote in yalmip:

i have sdpvar objects: Ew=sdpvar(24,1); Ein= sdpvar(24,1);Eb=sdpvar(24,1);x=sdpvar(24,1); 
and i want to add the condition(to constraints) if(Ew+Eb>=x) then Ein=0. i tried to do it with set(implies(Ew+Eb>=x, Ein==0 )) but i get warnings:

You have unbounded variables in IMPLIES leading to a lousy big-M relaxation. 

You have unbounded variables in IFF leading to a lousy big-M relaxation

i think that implies is only used with binvar, please do you know any other way to do this with sdpvar objects. thanks in advance

[Johan Löfberg]

In order to model linear constraints implies other linear constraints, binary variables have to be introduced internally. These models are created in YALMIP using so called big-M modelling.

http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Tutorials.Big-MAndConvexHulls

As a simple example, let us assume you have a binary variable d, and you have the constraint d==0 implies x==0 where x is a continuous variable. Let us also assume that you know that the optimal x is bounded in amplitude, for instance -10<=x<=10.

Then, a big-M model would be

    -10*d <= x <= 10*d

Right!?

To able to derive such a model, the magic value 10 is crucial. We must know a bound on the variable. The smaller it is, the better the model will be, but it must of course not cut off any part of our region of interest. Hence the term big-M (in theory, pick it to infinity, in practice large values kills the solver numerics so large amount of effort should be spent on deriving good values)

In your model, YALMIP has failed to find any such bounds on the variables that are involved in your logic constraint. You must add them.

Now, another issue is that your use of implies is perhaps not what you want. The implies operator works with the first argument as a whole, i.e., it is not vectorized. For the implication to be activated, all constraints in the first argument must be true. In other words, the first argument in your case is a polytope, and it is true if all constraints are true. If you want the logic that if the fifth row of the first argument is true, then the fifth element of Ei is zero, then you have to vectorize the code.

Some code to illustrate a vectorized vs non-vectorized setup

    

    % Logical constraints

    F = [];

    for i = 1:length(x)

        F = [F, implies(Ew(i)+Eb(i)>=x(i), Ein(i)==0)];

    end

    % Bounds required. Let us assume we know

    F = [F , -10 <= Ew <= 10,-20 <= Eb <= 20, -50 <= x <= 50, 0 <= Ein <= 100];

    % Now assume we have some model which leads to the first ten constraints

    % being satisfied! 

    F = [F, Ew(1:10)+Eb(1:10) >= 1, x(1:10) <= 0]

    % If we solve this model, the first 10 Ein elements have to be 0 

    % We can see this for instance if we try to force Ein to the point 2.

    % Despite trying to force Ein to 2, the first 10 stays at zero. Hence, the

    % model works

    solvesdp(F,norm(Ein-2))

    double(Ein)

    % A non-vectorized model on the other hand, would lead to all elements being

    % 2, since the constraint says if all elements are satisfied then Ein has 

    % to be zero. We are only forcing the first 10, so the solver is clever

    % enough and violates the remaining ones, thus turning off the implies

    F = implies(Ew+Eb>=x, Ein==0)

    F = [F , -10 <= Ew <= 10,-20 <= Eb <= 20, -50 <= x <= 50, 0 <= Ein <= 100];

    F = [F, Ew(1:10)+Eb(1:10) >= 1, x(1:10) <= 0]

    solvesdp(F,norm(Ein-2))

    double(Ein)

[nadiauir]

Dear Mr Lofberg,

Thnks so much for your help, i added the bounds as you said and i used the implies as you described it, i get no more warnings, but the problem is that the constraint in implies is not satisfied, i think because it is only appliable in the case of binary variables.

[Johan Löfberg]

Implies works with general linear constraints, as my example above shows if you run it (although implies with general linear constraints is an ill-posed model, but that is another story)

If you have a model in which this doesn't work, please supply that piece of code (as short and concise as possible), and point out in which sense the solution doesn't satisfy the constraint.