## 1 Introduction
Third, the _cardinality_ constraint, which imposes a floor and a ceiling on the number of assets included in the portfolio, accounts for the fact that diversification benefits decrease when the portfolio features a huge number of assets. When the above additional constraints are incorporated in the portfolio optimization problem, the constrained efficient frontier (CEF) can be obtained. In the presence of these rich constraints, the problem becomes NP-hard [31] and, thus, exact optimization methods quickly lose their efficiency as the number of considered assets grows. The cardinality constraint also implies that the mean-variance frontier can become discontinuous for certain values of expected return [32]. In summary, as pointed out by [33], the cardinality and quantity constraints make large-size instances of the problem to be computationally intractable using traditional optimization approaches. Because our research involves the CEF, we devise a matheuristic algorithm for rich portfolio optimization (ARPO) that is based on the combination of an iterated local search (ILS) metaheuristic [34; 35], quadratic programming, and biased randomization strategies [36; 37; 38]. The contribution of this study is fourfold. First, we show that using a carefully devised matheuristic solver can significantly reduce the minimum running time necessary to obtain near-optimal solutions. Indeed, as it will be discussed later, the computing-time performance of ARPO is significantly better than those of the solvers proposed by [39; 40; 41]. Second, in terms of the minimum average percentage loss, the CEF determined by ARPO is approximately as close (or even closer in most cases) to the unconstrained efficient frontier as in the aforementioned studies. Third, the adoption of an initial solution that is based on a well-chosen criterion makes our solver more flexible when solving tight instances. The rate of return on an individual asset is used as a main criterion to construct the initial solution. It is worth noting that this criterion provides the best possible solution in terms of guarantying feasibility of the required portfolio return -i.e., if this initial solution is not feasible, then the problem has no feasible solution. By contrast, in most of the existing approaches, an initial solution is randomly drawn and, hence, the feasibility of these initial solutions cannot be guaranteed. Fourth and foremost, our paper relaxes the widely held assumption that inputs (individual asset returns, variances and covariances) are accurately measured and deterministic. In this regard, we show that ARPO can be used to solve portfolio optimization problems, in which a number of scenarios comprising uncertain returns, variances and covariances are studied and compared with the CEF-ARPO with accurate inputs. As expected, the CEF-ARPO with uncertain individual asset returns manifests in a higher portfolio variance value for a given required rate of return. By contrast, no added uncertainty to individual asset returns translates into no material change to the CEF-ARPO with accurate inputs when variances and covariances between individual asset returns are subjected to a random disturbance. Finally, we further carry out a stability analysis--involving small variations in cardinality and quantity constraints--that shows no material deviations from the benchmark CEF in terms of the portfolio variance and computational time. The remainder of this article is organized as follows. Section 2 provides a review of the literature related to the theme, also discussing the limitations of traditional methods and the need for new approaches based on metaheuristics. Section 3 gives a formal description of the optimization problem being considered, while Section 4 provides an overview of the ARPO algorithm, including its pseudo-code for quick implementation. Section 5 introduces the numerical experiments performed to test the algorithm performance, while the results are discussed in Section 6.