202 Optimizing Optimization
computational restrictions. In fact, already in the 1950s Markowitz pondered
using downside semivariance as a measure for risk — but rejected it on mainly
computational grounds. Today still, standard optimization methods like linear
or quadratic programming allow to solve portfolio selection problems only
under simplifying assumptions. So altogether, the third question is actually
a very strong limit to the implementation and testing of alternative, possibly
superior models.
To give just one example (which we do not claim to be a superior model),
assume we wish to minimize the value at risk (VaR) of an equity portfolio.
Figure 9.1 shows the search space of such a problem, which is a direct map-
ping from different weight combinations to the portfolio’s VaR.
As can be seen, there are many local minima. Hence, standard methods,
which usually use gradient information, cannot be applied here, since they will
stop at the first local minimum.
There are different approaches to deal with such problems. Sometimes we
can reformulate the model until it can be solved with standard methods, in
particular linear programs; for example see Rockafellar and Uryasev (2000) ,
Chekhlov, Uryasev, and Zabarankin (2005) , or Mausser, Saunders, and Seco
(2006). In such cases, powerful solvers are available that can efficiently handle
even large-scale instances of given problems. But there is also a cost, namely
that the formulation is very problem-specific and often relies on simplifications
or specific assumptions. Passow (2005) for instance shows how to optimize
00.050.10.02 00.06 0.040.1 0.0899.29.49.69.810x 10−^3Weight of asset 2 Weight of asset 1Value-at-riskFigure 9.1 Search space for VaR.