ismember becomes higher-dimensional

[Marc Wijnand]

I have two 2D sets X,Y. The set Z is defined using these two sets, and although it physically represents a 2D set with variable z (cf. plot(Z,z)), it is [6D] in Yalmip. Therefore, it is not possible to use the set Z as constraint of a new 2D variable. What can I do?

CODE:

%given sets X,Y

X = Polyhedron([rand(2,2);-rand(2,2)],rand(4,1));

y = sdpvar(2,1);

Y = [y'*y<=1];

%calculate set Z = X+Y (Minkowski sum)

x = sdpvar(2,1);

z = sdpvar(2,1);

Z = [z==x+y ismember(x,X) ismember(y,Y)];

%plot(Z,z)

%use Z in constraints of optimization problem

t = sdpvar(2,1);

plot([ismember(t,Z)]); %error

Error using sdpvar/replace (line 37)

Both arguments must have same size

Error in lmi/replace (line 24)

    F.clauses{i}.data = replace(F.clauses{i}.data,X,W,expand);

Error in lmi/ismember (line 4)

F = replace(F,recover(depends(F)),x);

[Johan Löfberg]

Of course, you have explicitly constructed it as an object living in 6d (z,x,y). I would not be surprised if it is impossible to represent the Minkowski sum of a polytope and an ellipsoid as a constraint in 2d using e.g. an intersection of quadratic inequalities. perhaps there exist an sdp representation or something like that. The representation that t is in the minkowski sum is done just as you've done with z, i.e, the variable z is in the minkowski sum.