- The objectives of such problems may include: expectedreturn, standard deviation and variation coefficient of the portfolioreturn rate. - Especially, an updated version of the interactive fuzzy utility method, named UIFUM, is proposed to deal with portfolio multi-objective optimization problems.. - Keywords: Portfolio optimization, mathematical programming, single-objective optimization, multi-objective optimization, computational optimization methods.. - Modern portfolio theory, fathered by Harry Markowitz in the 1950s, assumes that an investor wants to maximize a portfolio's expected return contingent on any given amount of risk, with risk measured by the standard deviation of the portfolio's return rate. - In this paper, the classical PO problem is considered: There are k assets (stocks)for possible investment. - The PO problem is to choose the weights w 1 , w 2. - For the PO problem we need the notations:. - The other constraints are optional ones that may be included in the problem formulation depending on circumstances.. - If we choose to optimize only one objective out of the three as shown above, then we have a single-objective optimization problem. - If we choose to optimize at least two of the three objectives (or some other additional objectives), then we have a multi-objective optimization problems. - In the traditional, classical setting, when all the coefficients of the programing. - problem are real numbers, the PO problem is to be solved in the crisp environment (see [4-6]).. - In this modeling setting, the 2 nd constraint and the 3 rd constraint should be changed appropriately, and the programming problem thus obtained is to be solved in the stochastic environment (see [4-6]).. - We also can apply the fuzzy programming to model the objectives and the constraintsof the PO problem as the fuzzy goals and flexible constraints. - In other cases, one can use the fuzzy utility objectives to deal with the multi- objective nature of the problem. - In all these cases the resulting programming problemis to be solved in the fuzzy environment (see [4-6]).. - To get numerical solutions of the PO problem, appropriate commercial computing software packages or scientific computing software packages can be chosen.. - In the next section of the paper, section 2, some mathematical programming models of the PO problem will be reviewed. - Then, in section 3, a single-objective optimization model of the PO problem will be considered and solved in the crisp environment. - In section 4, some aspects of computing optima of the multi- objective optimization model of the PO problem will be discussed, especially an updated version of the interactive fuzzy utility method will be considered for the purpose.. - Some mathematical programming models of the PO problem. - Now, the mathematical programming model for the PO problem may be set as a stochastic programming problem:. - Problem 1 can be turned into a single- objective optimization problem in crisp environment as either of the following cases.. - Problem 1 can also be turned into the following three-objective optimization problem wherein the objectives are treated as fuzzy utility objectives in the fuzzy environment.. - If in the problem 1 we treat the 1 st objective as stochastic objective and other objectives as level constraints, then we have a single- objective optimization problem which is to be solved in the stochastic environment.. - Finally, problem 1 can be re-formulated as a two-objective optimization problem which is to be solved in the mixed fuzzy-stochastic environment.. - It should be mentioned here that in the literature on computing optima for the PO problem much attention is focused on the single-objective optimization models and very less attention is paid to the multi-objective optimization models in the fuzzy environment and stochastic environment (see [2, 3]).. - Computing the optimal solutions for the single-objective optimization model of the PO problem. - The problems 2a, 2b and 2c as stated in section 2 are all single-objective optimization problems. - contain at least one non-linear function either in the objective or in the constraints, where there is the expression:. - Illustrative example: There are 08 stocks with the return rates R i as given in the following table:. - The use of the RST2ANU computational software package (which was designed based on the RST2ANU method) with the initial guess point w provides the following numerical solutions:. - All these weight vectors give the same optimal value of the largest expected return rate of the portfolio. - providing the lowest standard deviation of the portfolio return rate. - providing the largest value of the inverse of the variation coefficient of the portfolio return rate:. - Some aspects of computing optima of the multi-objective optimization model of the PO problem. - We can update “the interactive fuzzy utility method” (IFUM method), which initially was proposed for solving multi-objective linear programming problems (see [4, 5]),to solve multi-objective nonlinear programming problems. - This updated version of the IFUM method is first time proposed in this paper (the updated version is named as UIFUM). - ii) Using the RST2ANU procedure to find out the optimal solutions for each of the (three) objectives subject to the given constraints. - The results are summarized in the pay-off table as follows:. - wherein W 1 , W 2 and W 3 are the optimal solutions of the (three) single-objective optimization problems, respectively.. - iv) The initial set of optimal solutions of the problem 3 is O p = {W 1 , W 2 , W 3 } containing (weak Pareto) optimal solutions.. - i) Specify positive values s 1, s 2 , s 3 for weights of the fuzzy utility functions which are chosen by the decision maker (DM) depending on his/her subjective judgment. - ii) Construct the aggregation utility objective function based on the values of the weights as specified above:. - P of the portfolio . - P of the portfolio VC P ) -1. - i) Using the RST2ANU procedure to find out the optimal solution of the obtained single- objective programming problem:. - After the termination, the set O p of optimal solutions corresponding to different weighting sets S = (s 1 , s 2 , s 3 ) may be summarized in the following table.. - P of the. - Based on the information of the above table, the DM may choose the most preferred optimal solution to implement his/her investment portfolio. - Invest 26.30% of the total fund into the 6 th stock (TLT), 29.53% into 7 th stock (LQD) and 44.17%. - It is interesting to note that the optimal solutions as summarized in the above table all belong to the set of Pareto optimal solutions (also called efficient solutions). - This set may be considered as the theoretical extension of the efficient frontier, which graphically represents the efficient portfolios obtained when only two first objectives out of the three are considered.. - This paper deals with some modeling and computing aspects of the classical PO problem. - It has been shown that the PO problem can be modeled as a single- objective or a multi- objective programming problem which may be, depending on the realistic circumstances, treated in a crisp, stochastic and / or fuzzy environment.. - Although the illustrative example is quite a classical and simple one, it has been indicated that the PO programming problem is not a linear programming and not necessarily to be a convex or d.c. - Because of this reason, the PO problem is challenging all the experts in the field of mathematical programming and computational optimization to find out the. - global optima or the best investment decisions of the PO problem.. - This paper has also shown that the RST2ANU method can be of use in computing optima for the PO single-objective as well as multi-objective programming problems. - An updated version of the interactive fuzzy utility method (IFUM) has been proposed first time in this paper to find the optima of the PO multi-objective programming problem. - Because of the time limitation, we could not show how to use the updated versions of multi-objective optimization methods (the reference direction interactive method, called RDIM, and the interactive satisficing method, named PRELIME which were developed by us, to solve the PO problem as has been formulated in section 2 (see Problem 4 and Problem 5).. - Therefore, the scope for further research in modeling and computing optima of the PO problem is, first of all, to improve the efficiency of the existing computational optimization methods, including all computational techniques as mentioned in this paper as well as some others.
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