# Why is it better to use evolutionary algorithms than OLS for solving index tracking problem?

I am currently using different optimization algorithms for finding constrained portfolio that best replicate choosen index. So i have a optimization task to minimize tracking error. I wonder why every paper use evolutionary algorithms, particle swarm optimizers or to lesser extent simmulated annealing or bayesian optimization, when using standard OLS with constrains should suffice as it is already minimizing similar measure analyticaly, ergo more preicisely. My comparison have also shown that PSO or bayesian optimization wont converge and gives worse recommended parameters than OLS. Why are they more popular ?

• Some people have problems to solve and they look for mathematical techniques to solve them. Some people know fancy mathematical techniques and look for problems to solve, so they can publish articles in journals showing how innovative they are in applying new techniques. – Alex C Aug 10 '19 at 23:02

## 1 Answer

It would have been helpful had you provided links to those papers.

But in general, you need to distinguish between the optimisation model, and the numerical technique used to solve the model.

Suppose you wanted to estimate a linear regression, with the mean squared residual as the criterion of fit, and without further constraints. This is a model. Now you can solve this model via a QR decomposition, say; or you can use a heuristic such as Differential Evolution or Particle Swarm Optimisation (PSO). If done properly, all techniques will give you exactly the same fit (up to numerical precision). That is because they all solve the same model, and it is a model that is easy to solve. (Btw, you cannot use Least-Squares techniques when you have inequality constraints.)

The advantage of using heuristics such as PSO is that you can now solve other, more complex models: you may, for instance, include cardinality constraints or UCITS (5/10/40) constraints. See for instance "The Threshold Accepting Heuristic for Index Tracking" or "Exact and Heuristic Approaches for the Index Tracking Problem with UCITS Constraints".

• Hi: Unless I'm mis-understanding, minimizing tracking error of portfolio versus fund is a quadratic optimization problem so I don't see how you could use OLS to solve it ? – mark leeds Aug 11 '19 at 8:31
• thanks for reply, maybe the constraints are what limits OLS. @markleeds you can use OLS to solve tracking error when y is index returns and explanatory variables are returns of assets choosen for portfolio. Estimeted model coefficients are weights of assets in a portfolio. Estimation of intercept must be ommited – SquintRook Aug 11 '19 at 9:47
• @SquintRook: Thanks but can you write out your formulation. The standard tracking error problem is quadratic ( return is linear but variance is quadratic ) so you must be formulating it differently than the standard approach. – mark leeds Aug 11 '19 at 18:41
• @mark leeds: in the appendix of enricoschumann.net/files/style_analysis.pdf, there are examples for unconstrained optimisation models that can be solved with a QP or a Least-Squares method. You could still use a Least-Squares method when you have linear equality restrictions (such as the sum of coefficients should be one), but not inequality restrictions (such as nonnegativity). – Enrico Schumann Aug 12 '19 at 6:29
• @Enrico Schumann: Thanks for your examples but what does $X$ represent in those two cases. Maybe they are return exposures to some underlying factor model ? So X represents the loadings that the target fund has with respect to that factor model ? Then, the resulting $\beta_{i}$ are the weights that should be used for each of the factors ? Thanks. – mark leeds Aug 12 '19 at 14:12