In practical applications of optimization it is common to have several conflicting objective functions to optimize. Frequently, these functions are subject to noise, or can be of black-box type, or highly nonsmooth, preventing the use of derivative-based techniques.
We propose a novel multiobjective derivative-free methodology, called direct multisearch (DMS). Our framework is inspired by the directional polling scheme of direct-search methods and uses the concept of Pareto dominance to maintain a list of nondominated points. DMS generalizes to multiobjective optimization all direct-search methods of such type. We prove under the common assumptions used in direct search for single objective optimization that at least one limit point of the sequence of iterates generated by DMS lies in (a stationary form of) the Pareto front. However, extensive computational experience has shown that DMS an impressive capability of generating the whole Pareto front.
Inspired on this work, we have developed a new methodology to deal with the computation of mean-variance Markowitz portfolios with pre-specified cardinalities. Instead of imposing a bound on the maximum cardinality or including a penalization or regularization term into the objective function (of classical Markowitz mean-variance models), we take the more direct approach of explicitly considering the cardinality as a separate (discontinuous) goal. This lead us to a cardinality/mean-variance biobjective optimization problem, whose solution is given in the form of an efficient frontier or Pareto front, thus allowing the investor to tradeoff among these two goals when having transaction costs and portfolio management in mind. For several data sets obtained from the FTSE 100 index and the Fama/French benchmark collection, direct multisearch (DMS) was capable of determining (in-sample) such efficient frontiers. Surprisingly, a significant portion of the efficient cardinality/mean-variance portfolios (with cardinality values considerably lower than the number of securities) have exhibited superior out-of-sample performance.
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