Does anyone know how to solve the following optimisation problem obtained from the paper "Spectral Risk Measures and Portfolio Selection" written by Alexandre Adam et al in 2007 :
Consider vectors π $\in R^d$. Let A be an $n$ by $d$ matrix and let $\lambda_i$ be constants such that $0<\lambda_1< \lambda_2 < .... < \lambda_n$.
Minimise $-\sum_{i=1}^{n} \lambda_i(Aπ)_{(i)} $ where $(Aπ)$ is a ($n$ by $1$) vector and $(Aπ)_{(1)}$ is the smallest component of $(Aπ)$, $(Aπ)_{(2)}$ is the second smallest component of $(Aπ)$, ..., $(Aπ)_{(n)}$ is the largest component of $(Aπ)$ subject to $\frac{1}{n}\sum_{i=1}^{n}(Aπ)_{(i)} = r $, some constant $r \in R$.
This is a very unusual problem because we have to order the components of the vector $Aπ$. The above is very general. I have tried to implement the ideas in the paper and got to the following more specific problem:
You have a function $f_D: R^{11} \rightarrow R$ such that for all $π \in R^{11}$, $f_D(π)$ is calculated as follows:
First you multiply the vector π by the (fixed) matrix A (which has dimension 92 by 11) to get another vector $A_π$ which has 92 components.
Then $f_D(π) = \frac{-5}{23}*(A_{{π}_{(1)}}+A_{{π}_{(2)}}+A_{{π}_{(3)}}+A_{{π}_{(4)}})-\frac{3}{23}*A_{{π}_{(5)}}$ where by $A_{{π}_{(1)}}$ I mean the smallest component in the vector $A_π$, $A_{{π}_{(2)}}$ is the second smallest component in the vector $A_π$, ...., $A_{{π}_{(92)}}$ is the largest component in the vector $A_π$.
I want to minimise $f_D(π)$ subject to the linear constraint that $π.\theta \geq 1$ where $\theta$ is a fixed vector.
In my case, $f_D(π)$ is a convex function and a solution definitely exists (by results taken from a paper written by Rockafellar called "Master Funds in Portfolio Analysis With General Deviation Measures") but when I tried using R I did not manage to find the soln.
Does anyone have any ideas on what I should do/any good convex optimisation functions in R?
Thank you in advance.