optimization proble risk parity

I state that I am not very experienced in MATLAB. I should create and minimize a function to find the weights of an asset allocation problem. The problem is as follow:

Minimize ∑(RC(Fj)/σP−1/M)^2 for j=1:M

sub ∑x=1

where RC(Fj) = (A⊤x)j⋅ (A+Σx/√x⊤Σx)j

where

• A is a loadings matrix
• A+ is Moore-Penrose inverse of A
• Σ is a covariance matrix
• x is a weights vector

I would need some codes to solve these problems. Thank you in advance.

Answer 1

[Rishik Ramena]

MATLAB does have a toolbox dedicated to solving optimization problems like these. Do have a look at its documentation for the ramp up.

[Tommaso Delicato]

I cannot extract the j-th component of the vector which is calculated as the product of the part (A+Σx/√x⊤Σx)j.

Can you tell me how?