Portfolio Optimization with LASSO

I have to find the optimal portfolio adding the "l-1 norm" constraint to the classical mean-variance model. How can i write this optimization in matricial form ?

Answer 1

[Ameer Hamza]

This shows an example for the case of 5 portfolios

mu = rand(1, 5);
eta = 0.5;
Sigma = ones(5);

Aeq = [mu; ones(1, 5)];
Beq = [eta; 1];

x0 = rand(5,1); % initial guess

sol = fmincon(@(x) x.'*Sigma*x, x0, [], [], Aeq, Beq, [], [], @nlcon);


function [c, ceq] = nlcon(x)
    c = sum(abs(x))-1;
    ceq = [];
end

[ANDREA MUZI]

I probably misplaced the question. I must obtain a portfolio, whose sum of the weights must be equal to 1, and the sum of the absolute value of the weights less than a certain t, so that when the t decreases some assets have weight equal to zero.

[Ameer Hamza]

You want the weighted sum to be equal to eta or less than eta? I have corrected a mistake in my code, and now it implements the constraints as written in the question.

[ANDREA MUZI]

equal to eta

[Ameer Hamza]

Then the code in my answer satisfies all the constraints. You can verify

mu = rand(1, 5);
eta = 0.5;
Sigma = ones(5);

Aeq = [mu; ones(1, 5)];
Beq = [eta; 1];

x0 = rand(5,1); % initial guess

sol = fmincon(@(x) x.'*Sigma*x, x0, [], [], Aeq, Beq, [], [], @nlcon);


function [c, ceq] = nlcon(x)
    c = sum(abs(x))-1;
    ceq = [];
end

Results

>> mu*sol % output is eta
ans =
    0.5000
>> sum(sol) % sum is 1
ans =
     1
>> sum(abs(sol)) % sum of absolute values is 1
ans =
     1

Answer 2

[ANDREA MUZI]

I thank you but it is not the result I expected; I try to rephrase the question. I found a way to linearize the constraint on the weights norm (photo). Basically I have to find the vector between tmin and tmax, in which tmin penalizes all the weights of the assets, bringing them to zero, except one whose weight will be equal to 1 and tmax, whose value will not penalize any asset