## 2 Need for New Metaheuristic- and Learning-Based Approaches  
An updated literature review on the portfolio optimization problem can be found in [2]. For this reason, our review focuses on analyzing the limitations of traditional approaches and discussing about the need of new algorithms that consider richer constraints and large-scale instances. Traditional optimization methods feature a number of theoretical and practical limitations. A notable characteristic of these methods is that they work only for problems that typically rely upon strict deterministic rules. First, they can produce wrong solutions when the portfolio selection problem under consideration has one or more local maxima in addition to a global maximum [8]. Second, the optimal solution may be disguised by the presence of estimation errors in expected returns and variances that are used as inputs [2; 42]. Third, the optimal solution is notoriously unstable, as small changes in inputs can cause large changes in the optimal portfolio weights [43]. Fourth, in terms of out-of-sample performance, traditional optimization methods are sometimes no better that the naive portfolio, wherein all assets are allocated the same weight [44]. Along similar lines, reference [45] argues that traditional optimization models can generate accurate in-sample forecasts, but their out-of-sample performance is generally weak. Fifth, a more realistic portfolio selection problem is typically confined to a subset of assets, which are selected by imposing additional constraints. A number of restrictions are specified in an investment management agreement between a client and a portfolio manager [2]. The client may impose a limit on the number of assets in the portfolio (_cardinality constraints_) [46]. Further, the client may also ask the manager to invest or not to invest in certain industries or companies. Therefore, the so-called _pre-assignment constraints_ may be used that pre-assign certain industries or assets in the portfolio. The client may further impose a discretionary limit on exposure to certain industries or assets that aims at keeping the quantity of each asset within a given range (_quantity constraints_). Another battery of constraints is dictated by the presence of _transaction costs_[2; 18]. If one or more of these constraints are introduced in portfolio optimization, indeterminacies in the efficient portfolio frontier may arise further limiting the usefulness of traditional optimization methods. In addition to the above theoretical limitations, traditional optimization methods have received a weak support in practice. Whilst diversification is a dominant portfolio selection strategy that contributes to lowering the risk of portfolio investment, in practice investors invest in fewer assets due to a variety of reasons. First, the administration of large portfolios can be cumbersome [41] or involve transaction costs in the form of bank and broker fees [47]. Second, there is evidence that investors' desire to diversify is limited [48; 49; 50]. Third, references [8; 22] argue that diversification can be achieved by investing in a small, yet well-chosen sub-set of assets. Similarly, reference [51] assert that, in most practical situations, investing in a portfolio comprising between 10 and 20 individual stocks can reasonably reduce the risk of investment. By the same token, [14] suggests including between 15 and 20 assets to diversify away most diversifiable risk. Reference [52] find that, quite often, the optimal diversification should not use more than 27 assets in real-life investment. Moreover, a smaller number of assets imply less number of parameters to be estimated by using sample information and, therefore, less room for estimation error [44]. Reference [42] provides a detailed account of advantages and disadvantages of traditional optimization methods. Specifically, he argues that these methods do not appeal to practitioners, since they do not make investment sense and do not have investment value.