## 1 Introduction  
Since [PERSON] [1], mean-variance optimization has become the workhorse model for portfolio selection. As discussed in [2], [PERSON]' s theory of portfolio selection has had a major impact on academic research and on the industry of financial services (Other recent studies on the theory of portfolio selection include [3, 4], among many others. Mean-variance portfolio optimization has also provided practical solutions to defined contribution pension schemes [5], hedging mechanisms [6], as well as to long-term fixed-income investments [7]). An important assumption underlying this model is that investors are concerned about both the expected returns from their investment and the risk from that investment, where risk is defined as the variance of future returns. By optimally allocatingappropriate weights to imperfectly correlated risky assets, investors can reduce the variance of future portfolio returns and thus diversify their investment. Because of the non-negativity constraint on the level of investment in each asset, a closed analytical solution is generally not feasible [8], and optimization methods need to be used to determine optimal portfolio weights. Since the mean-variance portfolio formulation, an enormous amount of papers have been published extending or modifying the basic model in three directions [9]. The first path goes to simplification of the amount or the type of input data [10]. The second path focuses on the introduction of an alternative measure of risk [11]. The third path involves incorporation of the additional criteria and constraints [12; 13]. Traditional optimization methods include, among others, linear programming [14; 15; 16; 17; 18; 19; 20], quadratic programming [21; 22; 23] and stochastic programming [24]. However, traditional optimization methods (notably, linear and quadratic programming) are plagued by a number of caveats. A notable characteristic of these methods is that they work only for problems that rely upon strict theoretical rules, albeit with a limited practical appeal [2]. Unfortunately, the complexity of the mean-variance model for portfolio optimization increases dramatically upon adding additional constraints that are needed to provide a rich representation of the investor's portfolio choice, which reduces the efficiency of exact methods in other than small-size instances. More recently, metaheuristic approaches have been proposed in the literature as a way to surpass these limitations [8; 25; 26; 27; 28]. Accordingly, this paper focuses on a single-period version of the so-called constrained mean-variance portfolio optimization problem. We will assume in this work that all assets are risky assets (stocks). However, the solving approach introduced here could be also used for similar optimization problems regarding commodities, futures, options and swaps. Moreover, as in [29; 30], our approach can be extended to include risk-free assets. The decision problem involves minimizing the portfolio variance for a given required rate of return. Portfolio weights add up to one and are constrained to take on non-negative values only. The latter assumption rules out short sales and thus places a constraint on excessive risk taking of investors. Under the above assumptions, the unconstrained efficient frontier (UEF) can be determined that gives, for each user-specified expected return, the minimum associated risk of investment. However, more realistic portfolio selection problems may involve additional constraints. First, justified on the grounds of the investor's preference and/or taste, the _pre-assignments_ force some specific assets to be included in the portfolio. Second, the _quantity_ constraint keeps the quantity of each selected asset within user-specified _floor_ and _ceiling_ values. The ceiling rules out excessive exposure to a specific asset. The floor is introduced in order to rule out the possibility of tiny (and therefore disproportionately costly) fractions of assets to be included in the portfolio.