MULTIOBJECTIVE EVOLUTIONARY METAHEURISTIC APPROACH TO THE CONSTRAINED PORTFOLIO OPTIMIZATION PROBLEM

Ikhlef, Massika; Aïder, Méziane

doi:10.1590/0101-7438.2023.043.00266962

Massika Ikhlef

LaROMaD, Fac. Maths, USTHB, Algiers, Algeria

Fac. Sciences, UMB-Boumerdes, Algeria - E-mail: m.ikhlef@univ-boumerdes.dz

http://orcid.org/0000-0002-3805-9716

Méziane Aïder ^**Corresponding author

LaROMaD, Fac. Maths, USTHB, PB 32 Bab Ezzouar, 16111 Algiers, Algeria - E-mail: meziane.aider@usthb.edu.dz

http://orcid.org/0000-0001-7195-810X

*Corresponding author

1: Step 0: Initialization of P-ACO.
2:			- Create Γ ants;
3:			- Initialize pheromone vectors with τ ₀;
4: Step 1: Iteration:
5: for each p ∈ P do
6:	- Fix the lifespan of the ant Ξ = K;				▷ K is the fixed number of assets to be selected;
7:	- Set y = (0,..., 0);			▷ Create an empty portfolio y for each ant, ∑_i y _i = 0 and \|y\| = n;
8:	- Set sum := 0 ; ▷ Create the variable sum which gives the number of assets included to y;
9:	- Determine, randomly from the interval [0, 1], the objective weight w _a for each objective a;
10:	while sum ≤ K do
11:		Select an asset i according to the probability distribution and add it to y;
12:		Update local pheromone information;
13:		sum := sum + 1;
14:	end while
15: end for
16: Step 2: Checking feasibility of portfolio y.
17: if portfolio y is feasible then
18:	check efficiency of portfolio y;
19: end if
20: if portfolio y is efficient then
21:	store portfolio y and remove dominated ones.
22: end if

1: for each solution x from MC do				▷ MC is the matrix containing all solutions obtained by applying the crossover operator;
2:	Calculate the objective value of the solution x;
3:	repeat
4:		- Set ε;			▷ a vector randomly chosen on the interval [0,1];
5:		- Calculate the new solution $x' = x + ε$ ;
6:		- Calculate the objective value of the new solution x ^′;
7:		ifx^′ is dominated by x then
8:			x: = x;
9:			repeat 4, 5, 6;
10:		end if
11:		ifx is dominated by x ^′; then
12:			repeat 4, 5, 6;
13:		end if
14:		ifx^′ and x are equivalent then
15:			savex^′ and x;
16:			x: = x^′;
17:			repeat 4, 5, 6;
18:		end if
19:	until termination criterion is true.
20: end for

SP 100 with 98 assets			SP 500 with 476 assets
K	Return	Risk	K	Return	Risk
50	0.213265	512.678372	10	0.1354794	0.1376716
	0.112512	422.242630		0.09525867	0.10177683
	0.190705	503.768738		0.08501251	0.08373711
	0.116.808	423.820148		0.06799201	0.03822819
	0.176744	497.690060		0.05084089	0.03679313
	0.171635	495.680658		0.04417696	0.03518582
	0.131665	429.933680		0.04118156	0.03308424
	0.135661	433.616562		0.02510106	0.02298574
	0.149273	433.901566		0.02734874	0.02596494
	0.157965	441.118896		0.02871795	0.0266791
	0.161059	443.883738	100	0.7540985	5.2578543
	0.164927	450.406884		0.4622423	3.2788443
	0.168713	456.368526		0.3448385	2.8524634
80	0.278026	1207.943510		0.2410349	2.3486477
	0.256027	1159.959632		0.6114916	4.1390230
	0.234959	1115.002572		0.4323154	3.2468762
	0.262656	1165.628104		0.3193307	2.6280283
	0.249885	1131.073054		0.547265	3.500980
	0.271474	1184.533588		0.050428	49.152482
	0.255227	1159.089108		0.2950339	2.4596917
	0.234178	1110.485858		0.5023697	3.3834249
	0.248012	1121.461844		0.3770722	2.8849828
	0.224832	1108.411672	300	1.561179	32.089310
	0.266096	1178.966284		1.174359	28.197236
	0.255227	1138.903924		0.9588885	23.8943005
	0.214503	1097.030830		1.415065	31.665333
				1.150750	25.919244
				1.389743	28.851918
				1.211427	28.791523

Nasdaq with 2196 assets			Hang Seng with 31 assets
K	Return	Risk	K	Return	28.791523	Risk
10	0.2080628	0.214315	10	0.058008	54.094946
	0.1503612	0.17065663		0.053634	50.604180
	0.1428687	0.1422794		0.4052092	3.1003410
	0.11326854	0.09698731
	0.09275083	0.08668862
	0.08419325	0.0847406
300	3.263283	55.474549
	2.021897	44.972060
	1.4616685	42.463714
	1.065910	32.841021
	0.7308886	29.8021161
	0.4063562	27.2410551
	2.270448	46.702699
	1.618908	43.829508

	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇	p ₈	p ₉	p10	p11	p12	p13	p14	p15	p16
port 1	0	0	0	1	1	0	0	1	1	0	0	1	0	0	0	0
port 4	0	0	0	0	1	0	0	1	1	0	0	1	0	0	0	1
port 7	1	0	0	0	1	0	0	1	1	0	0	1	0	0	0	0

	p17	p18	p19	p20		p21	p22	p23	p24	p25	p26	p27	p28	p29	p30	p31
port 1	0	0	1	1		0	0	1	0	0	1	0	0	1	0	0
port 4	0	0	1	1		0	0	1	0	0	1	0	0	1	0	0
port 7	1	0	1	1		0	0	0	0	0	1	0	0	1	0	0

	Return	Risk
port 1	0.058008	54.094946
port 4	0.053634	50.604180
port 7	0.050428	49.152482

\{\begin{array}{r} \max \sum_{i = 1}^{n} μ_{i} x_{i} & (4.1) \\ \min \sum_{i = 1}^{n} \sum_{j = 1}^{n} ρ_{i j} x_{i} x_{j} & (4.2) \\ \sum_{i = 1}^{n} x_{i} & = & 1 & (4.3) \\ \sum_{i = 1}^{n} y_{i} & = & K & (4.4) \\ y_{i} + y_{j} & \leq & 1 & \forall (i, j) \in F & (4.5) \\ x_{i} & \leq & y_{i} & \forall i \in E & (4.6) \\ x_{i} & \geq & 0 & \forall i \in E & (4.7) \\ y_{i} & \in & {0, 1} & \forall i \in E & (4.8) \end{array}

[1] *Corresponding author

Brasil

Brasil

MULTIOBJECTIVE EVOLUTIONARY METAHEURISTIC APPROACH TO THE CONSTRAINED PORTFOLIO OPTIMIZATION PROBLEM

ABSTRACT