A constrained consensus based optimization algorithm and its application to finance. (English) Zbl 1510.90208
Summary: In this paper, we propose a predictor-corrector type Consensus Based Optimization(CBO) algorithm on a convex feasible set. Our proposed algorithm generalizes the CBO algorithm in [S.-Y. Ha et al., Numer. Math. 147, No. 2, 255–282 (2021; Zbl 1467.65064)] to tackle a constrained optimization problem for the global minima of the non-convex function defined on a convex domain. As a practical application of the proposed algorithm, we study the portfolio optimization problem in finance. In this application, we introduce an objective function to choose the optimal weight on each asset in an asset-bundle, which yields the maximal expected returns given a certain level of risks. Simulation results show that our proposed predictor-corrector type model is successful in finding the optimal value.
MSC:
| 90C26 | Nonconvex programming, global optimization |
| 65K10 | Numerical optimization and variational techniques |
| 70F10 | \(n\)-body problems |
| 90C90 | Applications of mathematical programming |
| 91G60 | Numerical methods (including Monte Carlo methods) |
Citations:
Zbl 1467.65064References:
| [1] | Albi, G.; Bellomo, N.; Fermo, L.; Ha, S.-Y.; Pareschi, L.; Poyato, D.; Soler, J., Vehicular traffic, crowds, and swarms. on the kinetic theory approach towards research perspectives, Math. Models Methods Appl. Sci., 29, 1901-2005 (2019) · Zbl 1431.35211 |
| [2] | Bertsekas, D., Convex Analysis and Optimization (2003), Athena Scientific · Zbl 1140.90001 |
| [3] | Bottou, L., Online learning and stochastic approximations, On-line Learn. Neural Netw., 17, 142 (1988) |
| [4] | Carrillo, J. A.; Jin, S.; Li, L.; Zhu, Y., A consensus-based global optimization method for high dimensional machine learning problems, ESAIM: COCV, 27, S5 (2021) · Zbl 1480.60195 |
| [5] | Carrillo, J.; Choi, Y.-P.; Totzeck, C.; Tse, O., An analytical framework for consensus-based global optimization method, Math. Models Methods Appl. Sci., 28, 1037-1066 (2018) · Zbl 1397.35311 |
| [6] | Choi, Y.-P.; Ha, S.-Y.; Li, Z., Emergent dynamics of the Cucker-Smale flocking model and its variants, (Bellomo, N.; Degond, P.; Tadmor, E., Active Particles Vol.I Advances in Theory, Models, and Applications, Series: Modeling and Simulation in Science and Technology (2017), Birkhauser, Springer) |
| [7] | Eberhart, R.; Kennedy, J., Particle swarm optimization, Proc. IEEE Int. Conf.Neural Netw., 4, 1942-1948 (1995) |
| [8] | Gong, C.; Chen, H.; He, W.; Zhang, Z., Improved multi-objective clustering algorithm using particle swarm optimization, PLoS ONE, 12, 0188815 (2017) |
| [9] | S. Grassi, L. Pareschi, From particle swarm optimization to consensus based optimization: stochastic modeling and mean field limit, (2021) arXiv:2012.05613 · Zbl 1473.35570 |
| [10] | Ha, S.-Y.; Jin, S.; Kim, D., Convergence of a first-order consensus-based global optimization algorithm, Math. Models Methods Appl. Sci., 30, 2417-2444 (2020) · Zbl 1467.90040 |
| [11] | Ha, S.-Y.; Jin, S.; Kim, D., Convergence and error estimates for time-discrete consensus-based optimization algorithms, Numer. Math., 147, 255-282 (2021) · Zbl 1467.65064 |
| [12] | Holland, J. H., Genetic algorithms, Sci. Am., 267, 66-73 (1992) |
| [13] | van Laarhoven, P. J.M.; Aarts, E. H.L., Simulated Annealing: Theory and Applications (1987), D. Reidel Publishing Co.: D. Reidel Publishing Co. Dordrecht · Zbl 0643.65028 |
| [14] | Markowitz, H., Portfolio selection, J. Finance, 7, 77-91 (1952) |
| [15] | Nielsen, F., Introduction to HPC with MPI for Data Science (2016), Springer |
| [16] | Pinnau, R.; Totzeck, C.; Tse, O.; Martin, S., A consensus-based model for global optimization and its mean-field limit, Math. Models Methods Appl. Sci., 27, 183-204 (2017) · Zbl 1388.90098 |
| [17] | Poli, R.; Kennedy, J.; Blackwell, T., Particle swarm optimization: an overview, SWARM INTELL-US, 1, 33-57 (2007) |
| [18] | Totzeck, C.; Pinnau, R.; Blauth, S.; Schotthófer, S., A numerical comparison of consensus-based global optimization to other particle-based global optimization scheme, Proc. Appl. Math. Mech., 18 (2018) |
| [19] | Yang, X. S., Nature-Inspired Metaheuristic Algorithms (2010), Luniver Press |
| [20] | Yang, X.-S.; Deb, S.; Zhao, Y.-X.; Fong, S.; He, X., Swarm intelligence: past, present and future, Soft Comput., 22, 5923-5933 (2018) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
