A Minimax Portfolio Selection Rule with Linear Programming Solution
Management Science
Hardness of Approximating Problems on Cubic Graphs
CIAC '97 Proceedings of the Third Italian Conference on Algorithms and Complexity
Heuristic methods for the optimal statistic median problem
Computers and Operations Research
Hybrid Adaptive Large Neighborhood Search for the Optimal Statistic Median Problem
Computers and Operations Research
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We consider the following problem. A set $${r^1, r^2,\ldots , r^K \,{\in} \mathbf{R}^T}$$ of vectors is given. We want to find the convex combination $${z = \sum \lambda_j r^j}$$ such that the statistical median of z is maximum. In the application that we have in mind, $${r^j, j=1,\ldots,K}$$ are the historical return arrays of asset j and $${\lambda_j, j=1,\ldots,K}$$ 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.