Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

220 Optimizing Optimization

As a rule of thumb, the standard deviation of these accepted changes should be
of the same order of magnitude as the standard deviation of the changes in the

objective function values recorded from the random walk (i.e., the Δ (^) i -values in
Algorithm 3 or 4).
Figure 9.4 shows the thresholds for a given problem (with ΦDmean/1 )
as computed from the procedure given in Algorithm 4. The lower panel shows
the value of the objective function over time during an actual optimization
run. A vertical line indicates a switch to a lower threshold. Initially, the algo-
rithm moves freely, moving often to portfolios that map to higher (i.e., worse)
objective function values. Over time, as the thresholds decrease, the objective
function descends more smoothly. In the last few rounds, fewer portfolios are
accepted (as can be seen from the distance between the vertical lines), as the
algorithm becomes much more select.

9.6 Conclusion

In this chapter, we have outlined the use of a heuristic optimization method,
TA, for portfolio optimization problems. Heuristics are usually simple concep-
tually, but in our experience there is often quite a distance between a general
description of an algorithm and an actual implementation. We hope that this
chapter offers some helpful advice in this respect.

0 0.005 0.01 0.015 0.02

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000 6000 7000

0

0.1

0.2

0.3

Figure 9.4 A threshold sequence (upper panel) and the current value of the objective
function (lower panel).