We consider the following problem. A set r(1), r(2),..., r(K)is an element of R(T) of vectors is given. We want to find the convex combination z = Sigma lambda(j) r(j) such that the statistical median of z is maximum. In the application that we have in mind, r(j), j = 1, ..., K are the historical return arrays of asset j and lambda(j), j = 1, ..., K are the portfolio weights. Maximizing the median on a convex set of arrays is a continuous non-differentiable, non-concave optimization problem and it can be shown that the problem belongs to the APX-hard difficulty class. As a consequence, we are sure that no polynomial time algorithm can ever solve the model, unless P=NP. We propose an implicit enumeration algorithm, in which bounds on the objective function are calculated using continuous geometric properties of the median. Computational results are reported.
The optimal statistical median of a convex set of arrays.
RIZZI, ROMEO
2009
Abstract
We consider the following problem. A set r(1), r(2),..., r(K)is an element of R(T) of vectors is given. We want to find the convex combination z = Sigma lambda(j) r(j) such that the statistical median of z is maximum. In the application that we have in mind, r(j), j = 1, ..., K are the historical return arrays of asset j and lambda(j), j = 1, ..., K are the portfolio weights. Maximizing the median on a convex set of arrays is a continuous non-differentiable, non-concave optimization problem and it can be shown that the problem belongs to the APX-hard difficulty class. As a consequence, we are sure that no polynomial time algorithm can ever solve the model, unless P=NP. We propose an implicit enumeration algorithm, in which bounds on the objective function are calculated using continuous geometric properties of the median. Computational results are reported.
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