Solving Constrained Mean-Variance Portfolio Optimization Problems Using Spiral Optimization Algorithm
Abstract
:1. Introduction
2. Portfolio Optimization Problem as a Mixed Integer Nonlinear Programming Problem
2.1. Portfolio Optimization Problem
2.2. Mixed Integer Nonlinear Programming
3. Spiral Optimization Algorithm
Algorithm 1. Spiral Optimization Algorithm | |
1: | Input: |
, the number of search points maximum number of iterations | |
2: | Process: |
Step 1. Generate randomly initial points in the feasible region | |
Step 2. Set | |
Step 3. Find as | |
Step 4. Update to be | |
Step 5. Update to be Step 6. If then stop. Otherwise, set and return to step 4. | |
3: | Output: as a minimum point of |
4. Spiral Optimization Algorithm for the Portfolio Optimization Problem
Algorithm 2. Spiral Optimization Algorithm for Solving the MINLP Problem | |
1: | Input: |
, the number of search points ( maximum number of iterations | |
2: | Process: |
Step 1. Generate randomly initial points in the feasible region Step 2. Set Step 3. For , find where with | |
Step 4. Find as with Step 5. Update the value of to be such that we get () Step 6. Update the value of to be where with | |
Step 7. If then stop. Otherwise, set and return to step 5. Step 8. Set where | |
3: | Output : as minimum point |
Algorithm 3. SOA for Portfolio Optimization Problem with Buy-In Threshold Constraint | |
1: | Preliminaries: Spiral Input Parameter: m (≥2) the number of search points, θ (0 ≤ θ ≤ 2π), r (0 < r < 1), kmax maximum number of iterations Portfolio Input Parameter: Rp (target return), Q (covariance matrix), (penalty parameters), and (finite lower and upper bounds of assets proportion) |
2: | Initialization: |
Generate initial ,
, in the feasible search space randomly. | |
3: | Assign For i = 1,2,⋯, m, find where and with |
4: | Find and as and with |
5: | Update the value of to be and to be as follows: |
6: | Update the values of and and where with |
7: | If end to the program. Otherwise, choose and return to step 5. |
8: | Choose and |
9: | Output : and |
Algorithm 4. SOA for Portfolio Optimization Problem with Cardinality Constraint | |
1: | Preliminaries: Spiral Input Parameter: m (≥2) the number of search points, θ (0 ≤ θ ≤ 2π), r (0 < r < 1), kmax maximum number of iterations Portfolio Input Parameter: Rp (target return), Q (covariance matrix), (penalty parameters), and (finite lower and upper bounds of assets proportion) |
2: | Initialization: |
Generate initial , , in the feasible search space randomly. | |
3: | Assign For i = 1,2,⋯, m, find where and with |
4: | Find and as and with |
5: | Update the value of to be and to be as follows: |
6: | Update the values of and and where with |
7: | If then end the program. Otherwise, choose and return to step 5. |
8: | Choose and |
9: | Output : and |
5. Numerical Examples
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Stock Number | SOA MINLP Result Using Equation (7) with | Quasi-Newton in Bartholomew-Biggs | DIRECT in Bartholomew-Biggs | |
---|---|---|---|---|
1 | 0.1367 | 1 | 0.152 | 0.122 |
2 | 0.3893 | 1 | 0.381 | 0.369 |
3 | 0.3331 | 1 | 0.348 | 0.339 |
4 | 0.0911 | 1 | 0.118 | 0.111 |
5 | 0.0506 | 1 | 0 | 0.058 |
Risk | 0.6969 | 0.7005 | 0.6935 |
Stock Number | Cardinality Constraint Result Using Equation (8) with | |
---|---|---|
1 | 0.1288 | 1 |
2 | 0.3858 | 1 |
3 | 0.3267 | 1 |
4 | 0.0882 | 1 |
5 | 0.0675 | 1 |
Risk | 0.6978 |
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Febrianti, W.; Sidarto, K.A.; Sumarti, N. Solving Constrained Mean-Variance Portfolio Optimization Problems Using Spiral Optimization Algorithm. Int. J. Financial Stud. 2023, 11, 1. https://doi.org/10.3390/ijfs11010001
Febrianti W, Sidarto KA, Sumarti N. Solving Constrained Mean-Variance Portfolio Optimization Problems Using Spiral Optimization Algorithm. International Journal of Financial Studies. 2023; 11(1):1. https://doi.org/10.3390/ijfs11010001
Chicago/Turabian StyleFebrianti, Werry, Kuntjoro Adji Sidarto, and Novriana Sumarti. 2023. "Solving Constrained Mean-Variance Portfolio Optimization Problems Using Spiral Optimization Algorithm" International Journal of Financial Studies 11, no. 1: 1. https://doi.org/10.3390/ijfs11010001