Fireworks Algorithm Applied to Constrained Portfolio Optimization Problem

Transcription

1 Fireworks Algorithm Applied to Constrained Portfolio Optimization Problem Nebojsa Bacanin and Milan Tuba Faculty of Computer Science Megatrend University Belgrade Bulevar umetnosti 29, Belgrade, Serbia Abstract This paper presents implementation of the fireworks algorithm for portfolio optimization problem with constraints. Fireworks algorithm is a relatively new nature-inspired metaheuristic which emulates the process of fireworks explosion. We adapted fireworks algorithm for solving constrained portfolio selection problem that extends classical mean-variance portfolio model by adding additional constraints. Comparative analysis was conducted with three other swarm intelligence algorithms and three variants of genetic algorithm from the literature, using the same problem formulation and the same test data. Results show that the fireworks algorithm has great potential for tackling portfolio optimization problem since it performed better than mentioned algorithms considering all performance indicators. Keywords-portfolio optimization; fireworks algorithm; swarm intelligence; nature inspired algorithms I. INTRODUCTION Portfolio optimization (also known in the literature as portfolio selection problem), which deals with the optimal allocation of wealth among available assets, is one of the most important issues in economics and finance. Portfolio is collection of financial instruments (assets) owned by an individual or organization. Portfolios usually consist of bonds, stocks, futures, options, mutual fonds, etc. When making financial decisions, investors tend to maximize the return and to control the risk. These goals can be accomplished simultaneously by following the principle of diversification where investor tends to invest available capital into portfolio, rather than in a single asset. Portfolio optimization problem can be defined as multicriteria optimization problem in which risk has to be minimized, while return has to be maximized. However, the formulation of the problem in this way has several drawbacks [1]. It is difficult to collect enough data to make precise estimation of the risks and returns. Estimation of return and covariance (measure which models risk) is usually based on historical data and as such is error-prone. Finally, such problem formulation is simplification of the real-world environment because it does not capture many practical constraints, such as transaction costs, preferences over assets, maximum size of portfolio [1]. This research is supported by the Ministry of Education, Science and Technological Development of Republic of Srbia, Grant No. III /15/$31.00 c 2015 IEEE Portfolio optimization problem has been solved by using different methods and techniques like parametric quadratic programming technique [2] or linear programming [3]. However, as the model adopts additional constraints, it becomes harder for optimization and traditional, deterministic methods could not obtain satisfying results. In such cases, it is necessary to employ metaheuristic methods. This paper deals with the fireworks algorithm (FWA) that belongs to the group of swarm intelligence algorithms. Swarm intelligence mimics the behavior of animal colonies, such as flocks of bird schools of fish, colonies of ants or bees, populations of krills and bats, etc. It belongs to the group of population-based, iterative stochastic search techniques. The cornerstone of swarm intelligence is emergent behavior of many individuals that exhibits collective intelligence without central component that supervises the whole process. It is established on four basic properties on which self-organization rely: positive feedback, negative feedback, multiple interactions and fluctuations [4]. Swarm intelligence algorithms have been applied to the portfolio optimization problem. Particle swarm optimization (PSO) is one of the oldest swarm intelligence algorithms and its applications to different portfolio optimization models are presented in [5], [6]. Ant colony optimization (ACO) was used for solving Markowitz s mean-variance portfolio model [7], [8] and real estate portfolio optimization based on the average entropy [9]. Artificial bee colony (ABC) algorithm implementations for portfolio problem can also be found in the literature [10], [11]. Firefly algorithm (FA) which was devised by Yang [12] was also implemented for solving portfolio optimization [13], [14] and cardinality constrained mean-variance portfolio problem with entropy diversity constraint [15]. Krill herd (KH) is a relatively new nature-inspired metaheuristic devised by Gandomi and Alavi [16]. It was adapted for various tasks, including portfolio optimization [17]. In this paper, we present fireworks algorithm (FWA) adopted for solving constrained portfolio optimization problem. According to the available literature, FWA was not implemented for portfolio problem before. To prove the effectiveness of our approach, we performed tests using standard portfolio benchmark data from [18]. We also conducted comparative analysis with other optimization algorithms which were tested on the same benchmarks. 1242

2 The rest of the paper is organized as follows. In Section II we present mathematical formulations of different portfolio optimization models. Section III shows our implementations of the FWA. Optimization model, data sets, algorithm parameters, and empirical results along with the comparative analysis are given in Section IV. Finally, Section V concludes this paper. II. MATHEMATICAL FORMULATION OF PORTFOLIO OPTIMIZATION PROBLEM Portfolio optimization problem belongs to the group of multi-objective tasks since the goal is to maximize the returns while minimizing investors risk exposure. Thus, when making financial investment decisions, the investors tend to follow diversification strategy by investing the money into different types of assets. Many portfolio models exist in the literature, and in this section we show some of them. In the modern portfolio theory, one of the most important portfolio s models is the Markowitz s standard mean-variance model originally introduced more than fifty years ago [19]. In this model, the attractiveness of a security for investor is not only based on its arithmetic mean of return, but the standard deviation of the asset s return and its correlation with returns of other securities in the portfolio have also an important influence on the selection process [6]. In Markowitz s model, risk is defined as standard deviation of portfolio s return. The goal is to set a subset of assets from the predetermined pool of available assets in a way that the constructed portfolio generates minimum risk for a given level of return. When selection of risky portfolio is considered as the objective function, and the mean return of an asset is considered to be one of the constraints, the model can be defined as [6]: subject to min σ 2 R p = σ 2 p = R p = E(R p ) = j=1 ω i ω j Cov( R i Rj ) (1) ω i Ri R (2) ω i = 1 (3) ω i 0, i (1, 2,...N) (4) where N represents the number of potential assets, σp 2 is portfolio risk that should be minimized, Ri is the mean return on an asset i and Cov( R i Rj ) is covariance of returns of assets i and j respectively. Variable ω i is weight which determines the amount of the capital that is invested in asset i. The constraint formulated in Eq. (3) guarantees that the whole capital is invested. However, Markowitz s portfolio theory suffers from deficiencies and it has often been criticized mainly for the validity of variance as a risk measure. Besides that, it only provides a solution to asset allocation among the predetermined set of assets [20]. In the real-world investment environment, many different assets exist, such as bonds, options, stocks, future contracts, etc. The quality of these assets vary and it is very difficult task for the investors to find good quality assets because of information asymmetry and asset price fluctuations [20]. That is one of the reasons why this model can only be used as a starting point for building real word applicable models. Some portfolio methods model selection of assets by constructing only one evaluation function. These methods comprise of two main subsets: efficient frontier and Sharpe ratio models [5]. The goal of efficient frontier model is to find different objective function values by fluctuating desired mean return R. It is accomplished by introducing risk aversion indicator λ [0, 1]. This model can be written as [5]: min λ[ ω i ω j Cov( R i Rj ] (1 λ)[ ω i Ri ] (5) subject to j=1 ω i = 1 (6) ω i 0, i (1, 2,...N) (7) Risk aversion parameter λ sets the relative importance of the mean return to the risk. When λ increases, the relative importance of the risk increases for the investor, and the importance of the mean return decreases, and vice-versa [5]. Sharpe ratio (SR) model merges information from the mean and variance of an asset, and it can be formulated by Eq. (8) [21]: SR = R p R f StdDev(p), (8) where p denotes portfolio, R p is the mean return of the portfolio p, and R f is a test available rate of return on a riskfree asset. StdDev(p) is a measure of the risk in portfolio (standard deviation of R p ). One more model which is worth to be mentioned is Valueat-Risk (VaR). This model does not employ a variance as a measure of risk like in Markowitz s basic model, and it belongs to the group of quantile-based risk measures techniques. VaR describes the worst expected loss of a portfolio during a given time horizon under normal market conditions at a given confidence level. However, this model is represented with objective function that is very hard to be optimized, since it is non-linear, non-convex and non-differentiable [22]. All models presented so far are not realistic because they do not take into account all the constraints from the realworld environment. Some of the additional constraints include: budget, transaction lots, cardinality and sector capitalization constraints. 1243

3 Extended mean variance model that employs more realworld constraints was devised to make portfolio optimization more realistic. This model is classified as a quadratic mixedinteger programming task and can be mathematically formulated as [6]: where min σ 2 R p = σ 2 p = ω i = j=1 ω i ω j Cov( R i Rj ) (9) x i c i z i N j=1 x jc j z j, i = 1,..., N (10) z i = M N, M, N N, i = 1,...N, z i {0, 1} subject to where (11) x i c i z i Ri BR (12) x i c i z i B (13) 0 B lowi x i c i B upi B, i = 1,...N (14) W is W is i s i s y s y s 0, s, s {1,...S}, s < s (15) y s = { 1 if 0 if i s z i > 0 i s z i = 0, (16) where M is the number of selected assets from the pool of N available assets, B is the total budget for the investment, and B lowi and B upi are lower and upper limits of the budget that can be invested in asset i, respectively. Total number of sectors is represented as S, and c i represents the minimum transaction lot for asset i. x i denotes the number of c i that is purchased. Cardinality constraint is modeled with the decision variable z i which has a value of 1 if an asset i is included in portfolio, and if it is not, it has a value of 0. III. FIREWORKS ALGORITHM FOR PORTFOLIO OPTIMIZATION In this section we show our implementation of the fireworks algorithm (FWA) for constrained portfolio optimization problem. FWA is relatively new bio-inspired swarm intelligence approach devised in 2010 by Tan and Zhu [23]. It was inspired by the process of fireworks explosion. There are only few FWA s implementations that exist in the literature, but for those that exist, results show that it is a promising approach for both, bound-constrained and constrained problems. The very first paper that introduces the FWA reports results for bound-constrained global optimization that are better than the standard PSO (SPSO) and Clonal PSO (CPSO) [23]. In this work, FWA was tested on nine standard benchmarks. Later, many enhanced and upgraded FWA s for different kinds of optimization were presented and all of them proved to be robust methods by accomplishing outstanding performance [24]. There are also some hybrid approaches. For example, FWA was combined with the differential mutation (FWA-DM). This approach was successfully tested on CEC 2014 benchmark problems [25]. FWA was also tested on GPU (Graphical Processing Unit) in order to enhance computational performance. GPU-based Fireworks Algorithm (GPU-FWA) was implemented for solving large-scale problems [26]. Firework algorithm was inspired by the process of setting off a firework. When a firework is set off, a shower of sparks fill the local space around it. According to the opinion of the FWA s creators, the explosion process of a firework can be viewed as a search in the local space around a specific point where the firework is set off through the sparks generated in the explosion [23]. Firework explosion exhibits two characteristic behaviors. In well manufactured fireworks, numerous sparks are generated, and sparks tend to be in the center of the explosion. On the other hand, fireworks with lower quality manifest different behavior. In such cases, quite few sparks are generated, and the sparks are scattered in the space [23]. These two behaviors have implications on the search algorithm. In the first case, when a firework is well manufactured, a firework is located in the promising area of the search space which may be close to the optimal solution. Thus, it is better to generate more sparks to search around the firework. Contrarily, the second case, when sparks are scattered, implies that the optimal problem s solution may be far from the location of the firework, and that the search radious should be larger. In the FWA, more sparks are generated and the explosion amplitude is smaller for a good firework, compared to a bad one [23]. FWA belongs to the group of stochastic, iterative-based approaches. In each generation of explosions, n locations for the deployment of n fireworks are selected. After that, n other locations are selected from the current sparks and fireworks locations for the next iteration of execution. Basic cornerstones of the FWA are number of sparks and the amplitude of explosion. Let us suppose that the FWA is implemented for numerical minimization problem such as: min f(x), x = (x 1, x 2, x 3,..., x D ) S, (17) where x represents a real vector with D 1 components (parameters) and S R D is hyper-rectangular search space with D dimensions constrained by lower and upper bounds: lb i x i ub i, i [1, D]. (18) 1244

4 According to this formulation, the number of sparks which are created by each firework x i can be defined as [23]: s i = m y max f(x i ) + η n (y max f(x i )) + η, (19) where m represents a parameter that controls the overall number of sparks that are generated by the n fireworks, y max = max(f(x i ) (i = 1, 2,..., n) is the worst firework in the population (with the greates value of objective function in the case of minimization problems), and η is a small constant which is used to avoid division-by-zero error. It was concluded that satisfying results will not be obtained if s i is too big. So, it is necessary to define lower and upper bounds on s i according to [23]: round(α m) if s i < α m ŝ i = round(β m) if s i > β m, α < β < 1, (20) round(s i ) otherwise, where α and β are constant parameters. The second corner stone of the FWA is the amplitude of explosion. The amplitude of a well-designed firework is smaller than that in a bad one and this can be modeled with the following expression [23]: A i =  f(x i ) y min + η n (f(x i) y min ) + η, (21) where  represents the highest value of the explosion amplitude, while y min = min(f(x i ), i = 1, 2,..., n) denotes the best firework in the population of n fireworks (the one with the lowest value of objective function in the case of minimization problem). When an explosion occurs, z random dimensions (directions) of spark are affected. The number of affected dimensions is obtained by employing [23]: z = round(d χ), (22) where d is number of dimensions of optimization problem (the location x), and χ is a random number uniformly distributed between 0 and 1. In order to determine the location of a spark of the firework x i, a spark location ˆx j is generated first. The whole process is shown in the pseudo-code below [23]. Pseudo-code 1 Determine the initial spark s location: ˆx j = x i Select random z dimensions of ˆx j by using Eq. (22) Calculate the displacement: h = A i σ for each selected dimension ˆx j k of ˆx j do ˆx j k = ˆxj k + h if ˆx j k < xmin k map ˆx j k end if end for or ˆx j k > xmax k to the potential space In the presented pseudo-code, σ is a random number in the interval [ 1, 1]. For maintaining diversity of sparks (solutions in the population) a Gaussian method is employed. Gaussian(1, 1) function, which represents Gaussian distribution with mean 1 and standard deviation 1, is employed to define the coefficient of the explosion. Utilizing this method, ˆm sparks of this type are generated in each algorithm s iteration. Pseudo-code shows this kind of sparks initialization [23]. Pseudo-code 2 Determine the initial spark s location: ˆx j = x i Select random z dimensions of ˆx j by using Eq. (22) Calculate the coefficient of Gaussian explosion: g = Gaussian(1, 1) for each selected dimension ˆx j k of ˆx j do ˆx j k = ˆxj k g if ˆx j k < xmin k map ˆx j k end if end for or ˆx j k > xmax k to the potential space In each iteration, n fireworks locations are selected. Moreover, the current best location x is always kept and transfered to the next iteration so the algorithm uses elitism strategy. Later, n 1 locations are selected based on their distance to the other locations [23]. The distance between a position x i and other locations is calculated as: R(x i ) = j K d(x i, x j ) = j K x i x j, (23) where K is the set of all current locations of both fireworks and sparks. The probability of selecting location x i is calculated as [23]: p(x i ) = R(x i) R(x j ). (24) j K The process of algorithm execution can be summarized in the pseudo-code showed below. During each iteration, the algorithm generates two types of sparks according to Pseudocode 1 and Pseudo-code 2. The number of the first type sparks (denoted as m) and the amplitude of explosion is determined by the quality of the parent firework f(x i ). However, the number of the second type sparks, which are created from the Gaussian explosion, is fixed and controlled by the algorithm parameter ˆm. When algorithm obtains locations of both spark types, n locations are selected for the next iteration. FWA approximately performs n + m + ˆm function evaluations in each iteration [23]. Pseudo-code 3 Select random n locations for fireworks while maximum function evaluations is not reached do Set off n fireworks at n locations 1245

5 for each firework x i do Calculate the number of sparks for firework ŝ i by using Eq. (20) Obtain locations of ŝ i sparks of firework x i using Pseudo-code 1 end for for k = 1 : ˆm do Select random firework x j Generate a Gaussian spark for selected firework using Pseudo-code 2 end for Select best firework for next iteration Select randomly n 1 locations from the two types of sparks and the current fireworks with the probability given in Eq. (24) end while Appropriate constraint handling techniques has to be implemented to direct the search process towards the feasible domain of the search space. Equality constraints decrease efficiency of the search process by making the feasible space very small compared to the entire search space. In order to enhance the search process, the equality constraints can be replaced by inequality constraints using the following expression [27]: h(x) ε 0, (25) where ε > 0 is some small violation tolerance that is dynamically adjusted according to the current algorithm s iteration: ε(t + 1) = ε(t) dec, (26) In Eq. (26) t is the current iteration, and dec is a value slightly larger than 1. When the value of ε reaches the predetermined threshold value, Eq. (26) is no longer applied. To ensure that the sum of all assets weights included in the portfolio is equal to one, rather than treating it as a constrain, a different approach, the arrangement algorithm was adopted in [28]. This version of arrangement algorithm did not yield any improvements over the simplified version (for this special case when limits are 0 and 1 and desired sum is 1) that we first applied in the initialization phase of our FWA. We additionally modified it so that it includes handling the constraints from the Eq. (14) and Eq. (36): ˆω i = ω i D j=1 ω (1 D ω min ) + ω min, (27) j where ˆω i, i = 1, 2,..., D are adjusted values of weights ω i so that they always maintain feasibility. This modification significantly reduces the search space by more effectively dealing with equality constraints facilitating much better performance of the proposed FW algorithm. However, the approach used in [29] could introduce additional significant improvements because our simplified version maintains feasibility at the cost of rather significant disturbance in algorithm s search guidance. IV. PRACTICAL EXPERIMENTS In this section, we first show portfolio model and data set used for testing purposes. Later, we present test results and parameter adjustments for our implementation of the FWA. Finally, we show comparative analysis with genetic algorithms (GA) [18], firefly algorithm (FA) [13], artificial bee colony (ABC) optimization [30] and krill herd (KH) algorithm [17]. We note that all metaheuristics were tested on the same portfolio model using the same data set and number of function evaluations for a single run. A. Portfolio model and data set For testing purposes, we used portfolio model like in [18]. As shown in Section II, many portfolio models exist in the literature and most of them employ cardinality constraint, especially in the situation when the number of available assets is large. The model from [18] used here does not utilize the cardinality constraint since the number of available assets is 5 and this model is used only for initial testing and checking of the appropriateness of the new proposed method. The goal is to optimize weights of each asset in the portfolio in such way that the portfolio s return is maximized while, at the same time, the risk is minimized. To solve portfolio problem, a strategy of transforming multi-objective problem into single-objective one was used. The expected return of each individual security i in the portfolio is expressed as: E(ω i ) = w i r i, (28) where w i denotes the weight of individual asset i, and r i is the expected return of i. According to Eq. (28), the total expected return of the portfolio P can be defined as: E(P ) = E(ω i ), (29) where n is the number of securities in the portfolio P. Eq. (29) models first goal of the model, and that is to maximize portfolio s expected return. Second objective function that models portfolio s risk (variance) is defined as a polynomial expression of a second degree: σ 2 (P ) = ωi 2 σ 2 (r i ) + j=i+1 2ω i ω j Cov(r i, r j ), (30) where σ 2 (r i ) is the variance of asset i, and Cov(r i, r j ) is covariance between securities i and j. As stated before, an approach of transforming multiobjective problem into a single-objective was used, and thus, we combined Eq. (29) with Eq. (30) in a following way: max H(P ) = E(P ) σ 2 (P ) (31) Since in our experiments portfolio was defined as a minimization problem, we used the following expression: 1246

6 min H(P ) = σ 2 (P ) E(P ) (32) Alternatively, considering individual asset i, not the whole portfolio P, objective function can be formulated as: H(ω i ) = σ 2 (ω i ) E(ω i ) (33) In the paper [18] that we used for comparison, slightly different objective function that emphasizes expected return over the risk was used, so we reported the same quantity: Problem constraints are: min H(P ) = σ 2 (P ) + 1 E(P ) ω min i (34) ω 1 = 1 (35) ω i ω max i (36) r i ω i 0, (37) where ωi min and ωi max are minimum and maximum weights of asset i respectively. Constraint showed in Eq. (35) ensures that the all available assets will be included in portfolio, constraint in Eq. (36) defines lower and upper bounds for all assets, and Eq. (37) ensures that the positive portfolio s return is obtained. For testing purposes, we used simple historical data set from [18]. This data set is shown in Table I. It consists of the data for 5 stocks portfolio over a five years period. TABLE I DATA SET FOR THE EXPERIMENTS Year Stock 1 Stock 2 Stock 3 Stock 4 Stock We used this data set for easier comparison as a first testing phase of the FWA applied to portfolio optimization. On such a simple data set it is easy to see the behavior of the algorithm, how it handles constraints and how it converges. Next phase testing will be done on more complex benchmarks such as often used Hang Seng, DAX 100, FTSE 100, S P 100, and Nikkei 225 data set, but since they include large number of assets, inclusion of the cardinality constraint would be necessary, which would make direct comparison impossible. The mean return on each asset and covariance matrix are shown in Tables II and III respectively. TABLE II MEAN RETURNS FOR EACH ASSET Stock Stock Stock Stock Stock TABLE III COVARIANCE MATRIX Stock 1 Stock 2 Stock 3 Stock 4 Stock 5 Stock Stock Stock Stock Stock B. Experimental results and parameter settings To make the comparative analysis with other approaches more realistic, we used the total number of objective function evaluations of 240, 000, which considering the structure and parameters of our algorithm translates to 4,000 iterations. The same number of evaluations was used in tests with GA [18], FA [13], ABC [30] and KH [17]. Other FWA parameters are summarized in Table IV. TABLE IV FWA PARAMETER ADJUSTMENTS Parameter Value Number of fireworks (n) 5 Number of sparks 50 Constant parameter α 0.04 Constant parameter β 0.8 Highest value of the explosion amplitude  40 Number of Gaussian sparks ˆm 5 Initial violation tolerance (ε) 1.0 Decrement (dec) ω min 0.5 ω max 1 Since we used a set of five assets portfolio, problem dimension D was set to 5. Thus, each firework in the population is represented as a 5-dimensional vector. In the initialization phase, firework x is created using the following expression: x i = ω min i + rand(0, 1) (ω max i ω min i ), (38) where rand(0, 1) is a random number uniformly distributed between 0 and 1. All tests were performed on Intel CoreTM i7-4770hq with 32GB of RAM memory, Windows 8 x64 operating system and Visual Studio 2013 with.net Framework. As performance indicators, we used the best, mean and the worst results for objective function value. We also show risk (variance) and average return for all performance indicators. Algorithm was evaluated by 30 independent 1247

7 runs, each starting with a different random number seed. Testing results are showed in Table V. Results are rounded to 4 decimal places, but were calculated with grater precision. TABLE V EXPERIMENTAL RESULTS FOR 4,000 ITERATIONS Best Worst Mean St. Dev. Objective function Risk Return The same metric i.e. the best, the worst and mean value of objective function and corresponding portfolios variance and return averaged over 30 algorithm runs, was used in the paper with which we compare [18]. Standard deviation was not reported in [18] but we included it our results. As pointed out in Section IV-A, the objective function consists of two parts: variance and the return of portfolio, but for comparison purpose it is reported as combination defined by Eq. (34). In Table VI, we show portfolio asset weights of best and worst results. TABLE VI PORTFOLIO ASSET WEIGHTS FOR BEST AND WORST RESULTS Best Worst ω ω ω ω ω C. Comparative analysis with other approaches As already stated, we compared our algorithm with GA [18], FA [13], ABC [30] and KH [17] metaheuristics. In [18], three variants of genetic algorithms were shown: single-point, two-point and arithmetic. Arithmetic genetic algorithm obtains significantly better results than the first two variants. We included all three variants in the comparative analysis. Also, we should note that in [18], only the best result of objective function was shown, so, average and worst result of GA could not be included in the comparative analysis table. In Table VII, we made a comparative analysis with all approaches applied to the same portfolio model and same data set that could be found in the literature. The first paper where these test data were used is [18]. In the presented table GA SP represents genetic algorithm with single-point crossover, GA T P denotes genetic algorithm with twopoint crossover and GA A is arithmetic variant of genetic algorithm. For the sake of easier comparison, best results from each category are marked bold. From Table VII, we conclude that in overall, FWA performs better than other approaches. Best objective function result is better by 11.0 %, 4.1 %, 2.7 %, 2.9 %, 3.0 % and 3.4 % than the bests obtained by GA-SP, GA-TP, GA-A, FA, ABC TABLE VII COMPARATIVE ANALYSIS GA-SP GA-TP GA-A FA ABC KH FWA Best Worst N/A N/A N/A Mean N/A N/A N/A and KH, respectively. This proves significant advantage of the proposed approach over other metaheuristics. Similar conclusion could me carried out for the mean results comparisons, where FWA performs better by 3.5 %, 3.7 % and 4.8 % than FA, ABC and KH, respectively. Mean result for GA was not provided in the original research [18]. Thus, FWA also shows better average result than other state-of-theart algorithms. The same applies when comparing worst results, where FWA performs better by 4.2 %, 3.8 % and 4.1 % than FA, ABC and KH, respectively. It can be concluded that proposed approach is uniformly better and more robust. As mentioned before, we used 240,000 evaluations of objective function, which correlates to 4,000 iterations, in order to make fair comparison to [18], [13], [30] and [17]. However, we also tested the proposed algorithm with different numbers of iterations. It proved that FW algorithm is very robust, reaches acceptable solutions very quickly and continues to slowly and uniformly improve. The results for only 50 iterations are shown in Table VIII where it can be seen that these results are quite acceptable. TABLE VIII EXPERIMENTAL RESULTS FOR 50 ITERATIONS Best Worst Mean St. Dev. Objective function Risk Return The results for 50,000 iterations are shown in Table IX. Up to that point results continue to slowly improve where standard deviation becomes very small meaning that each run is practically perfect. Further increase of the number of iteration does not yield additional improvements for this precision of calculation. TABLE IX EXPERIMENTAL RESULTS FOR 50,000 ITERATIONS Best Worst Mean St. Dev. Objective function Risk Return V. CONCLUSION In this paper we described fireworks algorithm implementation for constrained portfolio optimization problem. The algorithm was tested on the standard benchmark portfolio that 1248

8 consists of five assets and historical data set from [18]. The results reported in this research prove that fireworks algorithm has great potential for tackling portfolio selection problem. The experiments were conducted by 30 independent runs each, performing 240,000 objective function evaluations. We compared our approach with three versions of the genetic algorithm, firefly algorithm, artificial bee colony algorithm and krill herd algorithm from the literature, which were all tested on the same data set and with the same number of function evaluations. As performance indicators, the best, the worst and mean values of objective function were used. By investigating experimental results, it can concluded that proposed fireworks algorithm implementation performs significantly better than other metaheuristics that were used for comparison. The results were uniformly better and robust yielding usable solutions in very few iterations and continuing to improve all the way to the precision of calculation. Considering the facts that there are many portfolio models that exist in the literature and that the fireworks algorithm has a great potential for tackling this kind of problem, further research could also be directed towards implementation of the fireworks algorithm for other portfolio models and testing it on larger data sets. REFERENCES [1] G. di Tollo and A. Roli, Metaheuristics for the portfolio selection problem, International Journal of Operations Research, vol. 15, no. 1, pp , [2] X.-L. Wu and Y.-K. Liu, Optimizing fuzzy portfolio selection problems by parametric quadratic programming, Fuzzy Optimization and Decision Making, vol. 11, pp , December [3] Y. Lan, X. Lv, and W. Zhang, A linear programming model of fuzzy portfolio selection problem, IEEE International Conference on Control and Automation, pp , May [4] E. 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