SHAMSHUL BAHAR YAAKOB et al: A HYBRID PARTICLE SWARM OPTIMIZATION APPROACH . . . IJSSST, Vol. 11, No. 5 ISSN: 1473-804x online, 1473-8031 print 68 A Hybrid Particle Swarm Optimization Approach To Mixed Integer Quadratic Programming For Portfolio Selection Problems Shamshul Bahar Yaakob 1,2 & Junzo Watada 1 1 Graduate School of Information, Production and Systems, 2-7, Hibikino, Wakamatsu, Kitakyushu, 808-0135 Japan 2 School of Electrical Systems Engineering, University Malaysia Perlis, 02600 Jejawi, Perlis, Malaysia shamshul@fuji.waseda.jp Abstract: Portfolio selection problems in investments are most studied in modern finance because of their computational intractability. The basic topic of modern portfolio theory is the way in which investors can construct a diversified portfolio of financial securities so as to achieve improved tradeoffs between risk and return. In this paper, a heuristic algorithm using particle swarm optimization (PSO) is applied to the problem. PSO realizes the search algorithm by combining a local search method through self-experience with global search method through neighboring experience, attempting to balance the exploration- exploitation trade-off which achieves the efficiency and accuracy of an optimization. A newly obtained effect is proposed in this paper by adding the mutation operator of genetic algorithms (GA) to unravel the stagnation and control the velocity. We applied our adaptation and implementation of the PSO search strategy to the portfolio selection problem. Results on typical applications demonstrate that the velocity information and mutation operator play pivotal roles in searching for the best solution, and that our method is a viable approach for the portfolio selection problem. Keywords: portfolio selection problem, particle swarm optimization, mean-variance approach. I. INTRODUCTION Portfolio selection problem consists in selecting a portfolio of assets or securities that provides the investor a given expected return and minimizes the risk. The principle of modern portfolio theory is that there could be an improvement in the tradeoff between risk and return, based upon the composition of any diversified portfolio of financial securities. By definition, portfolio optimization is a method to formulate the combination that can best achieve the objectives of the portfolio manager. The quadratic programming model of Markowitz [1] can be considered as one of the most established mathematical programming methods to solve the portfolio management problem. Based on this method, variance of return (a quadratic function) is used to represent the risk, which needs to be minimized subject to the achievement of a minimum expected return on investment (a linear constraint). On the other hand, Luenberger [2] has precisely explained the single-period model where the inputs of the analysis, such as security expected returns, variance and covariance for each pair of securities, are estimated from previous performance of the securities. Board and Sutcliffe [3] surveyed the application of conventional mathematical methods that have been used to model financial markets. Among these methods are linear programming, integer linear programming, nonlinear programming and dynamic programming models, all of which have been widely used to solve modern portfolio problems. When the model involves constraints on minimum trading quantities or on the maximum number of assets in the portfolio, as in the portfolio selection model, then we moves to the field of mixed integer nonlinear programming, in which classical algorithms are typically unable to deliver the optimum value of the problem. Actually, very few commercial packages are even able to handle this class of problems. Perold [4], whose work is most often cited in this context, included a broad class of constraints in his model, but did not place any limitation on the number of assets in the portfolio. His optimization approach is explicitly restricted to the consideration of factorial models, which, while reducing the number of decision variables, lead to other numerical and statistical difficulties due to the prerequisite of the proposed algorithm that employs a multi-factor approach to reduce the rank of covariance matrix and applies a quadratic programming algorithm. In order to specifically address the investment objectives of any individuals, corporations and financial firms, several portfolio strategies have been proposed. In this respect, the optimization strategy is selected according to the owner’s investment objective. Jones [5] has proposed a framework to classify alternative investment objectives. Some heuristic algorithms are proposed for the portfolio selection problem to extend the conventional approach, but it is not easy that conventional optimization