# Generate data for worst-case risk analysis.
srand(2);
n = 5;
r = abs(randn(n, 1))/15;
Sigma = 0.9 * rand(n, n) - 0.15;
Sigma_nom = Sigma' * Sigma;
Sigma_nom -= (maximum(Sigma_nom) - 0.9)
5x5 Array{Float64,2}: 0.54178 0.45192 0.328892 0.0123621 0.400576 0.45192 0.694173 0.416407 -0.237798 0.295939 0.328892 0.416407 0.496781 0.0266768 0.31054 0.0123621 -0.237798 0.0266768 0.414028 -0.181312 0.400576 0.295939 0.31054 -0.181312 0.9
# Form and solve portfolio optimization problem.
# Here we minimize risk while requiring a 0.1 return.
using Convex, SCS
set_default_solver(SCSSolver(verbose=0));
w = Variable(n);
ret = dot(r, w);
risk = sum(quad_form(w, Sigma_nom));
problem = minimize(risk, [sum(w) == 1, ret >= 0.1, norm(w, 1) <= 2])
solve!(problem);
wval = vec(evaluate(w))
5-element Array{Float64,1}: -0.258385 0.53885 -0.241623 0.751627 0.209527
# Form and solve worst-case risk analysis problem.
Sigma = Semidefinite(n);
Delta = Variable(n, n);
risk = sum(quad_form(wval, Sigma));
problem = maximize(risk, [Sigma == Sigma_nom + Delta,
diag(Delta) == 0,
abs(Delta) <= 0.2,
Delta == Delta']);
solve!(problem);
println("standard deviation = ", round(sqrt(wval' * Sigma_nom * wval), 2));
println("worst-case standard deviation = ", round(sqrt(evaluate(risk)), 2));
println("worst-case Delta = ");
println(round(evaluate(Delta), 2));
standard deviation = [0.27] worst-case standard deviation = 0.82 worst-case Delta = [-0.0 -0.2 0.19 -0.2 -0.2 -0.2 0.0 -0.2 0.2 0.2 0.19 -0.2 -0.0 -0.2 -0.2 -0.2 0.2 -0.2 0.0 0.2 -0.2 0.2 -0.2 0.2 0.0]