Lagrange Optimization w.r.t. vector + equality constraint

nikampe

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Nov 30, 2021
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Hi all,

I am currently facing the following optimization problem:

f(v)=rf+vμ+vdiag(Σ)1/2vΣv1/2+(1γ)vΣv1/2f(v) = r_f + v'*\mu + v'*diag(\Sigma)*1/2 -v'*\Sigma * v*1/2+(1-\gamma)*v'*\Sigma*v*1/2

s.t.

1v=11'*v=1 (sum of the vector elements = 1)

where Σ\Sigma = nxn variance-covariance-matrix, diag(Σ)diag(\Sigma) = nx1 variance vector, μ,γ,rf\mu, \gamma, r_f = constants, 11' = transpose nx1 vector of ones and finally the variable of interest vv = nx1 vector with v1,...,vnv_1, ..., v_n.

I defined the Lagrangian functions as L=f(v)+λ(1v1)L = f(v) + \lambda*(1'*v-1) and took the derivates w.r.t. vv and w.r.t λ\lambda. That's the point where I am stuck as I am not able to solve for lambda from the two functions and afterwards for the variable of interest vv. The goal in the end is to find the maximizing vector vv under the above equality constraint.

Could anybody help on how to solve this problem?

Thank you very much in advance.
 
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