Combining nonconvex MILP operator and uncertainty

[johanlofberg]

% Nominal case without any uncertainty
x = sdpvar(2,1);
Model = [-3 <= x <= 3, norm(x - [0.1;0.2],1) >= 1];
optimize(Model, x'*x)
clf
plot(Model);hold on;plot(value(x(1)),value(x(2)),'*')

% Add uncertainty, fails due to infeasibility
x = sdpvar(2,1);
w = sdpvar(2,1);
Model = [-3 <= x <= 3, norm(x + w - [0.1;0.2],1) >= 1];
Model = [Model, -0.5 <= w <= 0.5,uncertain(w)]
optimize(Model, x'*x)
clf
plot(Model);hold on;plot(value(x(1)),value(x(2)),'*')

% Manual derivation of nonconvex constraint 
% and it works as expected
x = sdpvar(2,1);
w = sdpvar(2,1);
d = binvar(4,1);
q = x + w - [0.1;0.2];
M = 10;
Model = [-3 <= x <= 3, q(1) + q(2) >= 1-M*(1-d(1)),
                       q(1) - q(2) >= 1-M*(1-d(2)),
                      -q(1) + q(2) >= 1-M*(1-d(3)),
                      -q(1) - q(2) >= 1-M*(1-d(4)),
                      sum(d)==1]
Model = [Model, -0.5 <= w <= 0.5,uncertain(w)]
optimize(Model, x'*x)
clf
plot(Model,x);hold on;plot(value(x(1)),value(x(2)),'*')

% Semimanual, represent the polytope
A = [1 1;-1 1;1 -1;-1 -1];
b = [1;1;1;1];
x = sdpvar(2,1);
w = sdpvar(2,1);
d = binvar(4,1);
q = A*(x + w - [0.1;0.2]) - 1;
Model = [-3 <= x <= 3, implies(d,q>=0),sum(d)==1];
Model = [Model, -0.5 <= w <= 0.5,uncertain(w)]
optimize(Model, x'*x)
clf
plot(Model,x);hold on;plot(value(x(1)),value(x(2)),'*')