Robust Markowitz Modell: Problems in the Covariance Matrix

[Lissi]

I'm still trying to implement a robust version of the markowitz modell. Thanks to you for helping me the last time!
I'm sorry, but now I have a new problem and I'm helpless, as it seems that its just a small alteration of the code.

Now I am trying to increase the uncertainties in the covariance matrix. First I just considered the variances Sigma(t) as uncertain, but now I want the covariances to be uncertain too. I calculate them as covariance(1,2) = Corelation(1,2)* sqrt(Sigma(1)*Sigma(2)) for example. But when I run the code, I get the error that my matrix is singular or badly scaled?
Do I have too many uncertainties or is my matrix just badly chosen?

Thank you!

% test data
% vector of return
mu_t = [0.05; 0.11; 0.16; 0.25];

% corelation matrix
Cor_t =     [1.0 0.1 0.7 -0.1;
            0.1 1.0 -0.2 0.4;
            0.7 -0.2 1.0 0.3;
            -0.1 0.4 0.3 1.0];

% Covariance matrix       
Sigma_t = [ 0.025 0.00055 0.0056 0.00125;
            0.00055 0.0121 -0.00352 0.011;
            0.0056 -0.00352 0.0256 0.012;
            0.00125 0.011 0.012 0.0625];

ab = 0.01;
lambda = 1;


% upper and lower bound for the box constraint of the vector of return
mu_l = mu_t - ab*mu_t;
mu_u = mu_t + ab*mu_t;

% upper and lower bound for the box constraint of the covariance matrix
Sigma_l = Sigma_t - ab*Sigma_t;
Sigma_u = Sigma_t + ab*Sigma_t;

n = length(mu_t);
w = sdpvar(n,1);
mu = sdpvar(n,1);


% Definition of Sigma, and the covariances, later to be uncertain
sdpvar t1 t2 t3 t4 t12 t13 t14 t23 t24 t34

t12 = Cor_t(1,2)*sqrt(t1*t2);
t13 = Cor_t(1,3)*sqrt(t1*t3);
t14 = Cor_t(1,4)*sqrt(t1*t4);
t23 = Cor_t(2,3)*sqrt(t2*t3);
t24 = Cor_t(2,4)*sqrt(t2*t4);
t34 = Cor_t(3,4)*sqrt(t3*t4);

Sigma =     [t1, t12, t13, t14;
            t12, t2, t23, t24;
            t13, t23, t3, t34;
            t14, t24, t34, t4];


% constraints on the portfolio
W = [sum(w) == 1, w >= 0];

% uncertainty for the vector of returns
M = [mu_l <= mu <= mu_u, uncertain(mu)];


% uncertainty for the variances
S = [Sigma_l(1,1)<= t1 <= Sigma_u(1,1),
    Sigma_l(2,2)<= t2 <= Sigma_u(2,2),
    Sigma_l(3,3)<= t3 <= Sigma_u(3,3),
    Sigma_l(4,4)<= t4 <= Sigma_u(4,4),
    uncertain(t1), uncertain(t2), uncertain(t3), uncertain(t4)];

% objective function
objective = lambda*w'*Sigma*w-mu'*w;
sdpvar t

solvesdp([W + S + M,lambda*w'*Sigma*w-mu'*w<=t] , t)

double(w)

[Johan Löfberg]

I am surprised the code even runs through the robustification. That uncertainty does not look conic representable. I think YALMIP doing something very wrong here to begin with.

[Lissi]

Thank you very much!  I've now changed the code to a SOCP. Anyhow, this time I get the error, that "Sigmonial SOCP not supported". What does that mean?

My code now is as follows:

% test data
 mu_t = [0.05; 0.11; 0.16; 0.16];

 Cor_t =     [1.0 0.1 0.7 -0.1;
             0.1 1.0 -0.2 0.4;
             0.7 -0.2 1.0 0.3;
             -0.1 0.4 0.3 1.0];
        

 Sigma_t = [ 0.025 0.00055 0.0056 0.00125;
             0.00055 0.0121 -0.00352 0.011;
             0.0056 -0.00352 0.0256 0.012;

             0.00125 0.011 0.012 0.027];
       

ab = 0.05;
R = mean(mu_t);

% upper and lower bounds for mu

mu_l = mu_t - ab*mu_t;
mu_u = mu_t + ab*mu_t;

mu_l(3) = 0.08;

% upper and lower bounds for Sigma

Sigma_l = Sigma_t - ab*Sigma_t;
Sigma_u = Sigma_t + ab*Sigma_t;

% upper and lower bounds for the corelation matrix
Cor_l = Cor_t - ab*Cor_t;
Cor_u = Cor_t + ab*Cor_t;

% definition of the variables
n = length(mu_t);

mu = sdpvar(n,1); %mu, later to be uncertain
w = sdpvar(n,1); 
Sigma = sdpvar(n,n); %Sigma, later to be uncertain

sdpvar z;           % additional variable

        % constraints on the portfolio
        W = [sum(w) == 1, w >= 0];

        % Uncertainty of mu
        M = [mu_l <= mu <= mu_u, mu'*w >= R, uncertain(mu)];

        %Uncertainty of Sigma, now it depends on the corelation matrix
        C = [];
        for k = 1:n
            for j = 1:n
                C = [C, norm([2*z;Sigma(k,k) - Sigma(j,j)],2) <= Sigma(k,k) + Sigma(j,j),
                Cor_l(k,j)*z >= Sigma(k,j) >= Cor_u(k,j)*z];
            end
        end

        for k = 1:n
            for j = 1:n
                C = [C, uncertain(Sigma(k,j))];
            end
        end
       
        %here I think lies the problem: I want to add a constraint on the risk Sigma, it shouldn't be any larger than a fixed value
        sdpvar y;
        y = Sigma.^(1/2)*w;
        C = [C, norm(y,2)<= 0.015];
     
       
        % Zielfunktion
        sdpvar t
     solvesdp([W + M + C,w'*Sigma*w<=t] , t)



Thany you very much in advance!

[Johan Löfberg]

You cannot have square roots of uncertainties etc and work with them arbitrarily. The only uncertainty scenarios supported are those listed in my paper Automatic Robust Optimization

You want w'*Sigma*w <= 0.015^2. Doesn't that work?

[Lissi]

it's working! I can't understand how I missed the totally obvious.
Thank you again!