Global Order-Value Optimization by means of a Multistart Harmonic Oscillator Tunneling Strategy

Summary

The OVO (Order-Value Optimization) problem consists in the min-imization of the Order-Value function f(x), defined by $f\left( x \right) = f_{i_p \left( x \right)} \left( x \right)$, where $f_{i_1 \left( x \right)} \left( x \right) \leqslant ... \leqslant f_{i_m \left( x \right)} \left( x \right)$. The functions f₁,..., f_m are defined on Ω ⊂ ℝⁿ and p is an integer between 1 and m. When x is a vector of portfolio positions and f_i(x) is the predicted loss under the scenario i, the Order-Value function is the discrete Value-at-Risk (VaR) function, which is largely used in risk evaluations. The OVO problem is continuous but nonsmooth and, usually, has many local minimizers. A local method with guaranteed convergence to points that satisfy an optimality condi-tion was recently introduced by Andreani, Dunder and Martinez. The local method must be complemented with a global minimization strategy in order to be effective when m is large. A global optimization method is defined where local minimizations are improved by a tunneling strategy based on the harmonic oscillator initial value problem. It will be proved that the solution of this initial value problem is a smooth and dense trajectory if Ω is a box. An application of OVO to the problem of find-ing hidden patterns in data sets that contain many errors is described. Challenging numerical experiments are presented.