longshort.m

%% Setup
n = 20;
x = sdpvar(n,1);
y = randn(n,1);
Objective = (x-y)'*(x-y);

%% Manual implies
d = binvar(n,1)
M = 1;
Model = [x >= -(1-d)*M, sum(d) >= 0.5*n, -1 <= x <= 1];
optimize(Model,Objective)

%% Built-in implies
Model = [implies(d,x>=0), sum(d) >= 0.5*n,  -1 <= x <= 1];
optimize(Model,Objective)

%% Sort
sorted = sort(x);
Model = [sorted(n/2:end) >= 0,  -1 <= x <= 1];
optimize(Model,Objective)

%% Median
Model = [median(x) >= 0,  -1 <= x <= 1];
optimize(Model,Objective)

%% Sign
Model = [sum(sign(x)) >= 0,  -1 <= x <= 1];
optimize(Model,Objective)

%% Nnz
Model = [nnz(x >= 0) >= n/2,  -1 <= x <= 1];
optimize(Model,Objective)