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
2009-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.