The OVO (Order-Value Optimization) problem consists in the minimization of the order-value function $F_p(x)$, defined by
F_p(x) = f_{i_p(x)}(x),
where
f_{i_1(x)}(x) \leq \ldots \leq f_{i_m(x)}(x).
The functions $f_1, \ldots , f_m$ are defined on $\Omega \subset 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. A Cauchy-like method with guaranteed convergence to points that satisfy a first order optimality condition was recently introduced by Andreani, Dunder and Mart\'{\i}nez. In this paper a quasi-Newton method is introduced that generalizes the Cauchy method. Convergence proofs are given. The new method is applied to a Brazilian-oriented variation of Stiglitz's Ideal Banking System.
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