Constrained Portfolio Optimisation: The State-of-the-Art Markowitz Models

Yan Jin, Rong Qu, Jason Atkin


This paper studies the state-of-art constrained portfolio optimization models, using exact solver to identify the optimal solutions or lower bound for the benchmark instances at the OR-library with extended constraints. The effects of pre-assignment, round-lot, and class constraints based on the quantity and cardinality constrained Markowitz model are firstly investigated to gain insights of increased problem difficulty, followed by the analysis of various constraint settings including those mostly studied in the literature. The study aims to provide useful guidance for future investigations in computational algorithms.


  1. Anagnostopoulos, K. and Mamanis, G. (2010). A portfolio optimization model with three objectives and discrete variables. Computers & Operations Research, 37(7):1285 - 1297. Algorithmic and Computational Methods in Retrial Queues.
  2. Anagnostopoulos, K. and Mamanis, G. (2011a). The meanvariance cardinality constrained portfolio optimization problem: An experimental evaluation of vfie multiobjective evolutionary algorithms. Expert Systems with Applications, 38(11):14208 - 14217.
  3. Anagnostopoulos, K. P. and Mamanis, G. (2011b). Multiobjective evolutionary algorithms for complex portfolio optimization problems. Computational Management Science, 8(3):259-279.
  4. Beasley, J. E. (1990). Or-library: Distributing test problems by electronic mail. Journal of the Operational Research Society, 41:1069-1072.
  5. Bienstock, D. (1995). Computational study of a family of mixed-integer quadratic programming problems. In Integer Programming and Combinatorial Optimization, volume 920 of Lecture Notes in Computer Science, pages 80-94. Springer Berlin Heidelberg.
  6. Bonami, P. and Lejeune, M. A. (2009). An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Operations Research, 57(3):650-670.
  7. Borchers, B. and Mitchell, J. E. (1997). A computational comparison of branch and bound and outer approximation algorithms for 01 mixed integer nonlinear programs. Computers & Operations Research, 24(8):699 - 701.
  8. Chang, T.-J., Meade, N., Beasley, J., and Sharaiha, Y. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13):1271 - 1302.
  9. Chen, A., Liang, Y.-C., and Liu, C.-C. (2012). An artificial bee colony algorithm for the cardinalityconstrained portfolio optimization problems. In 2012 IEEE Congress on Evolutionary Computation (CEC), pages 1-8.
  10. Cura, T. (2009). Particle swarm optimization approach to portfolio optimization. Nonlinear Analysis: Real World Applications, 10(4):2396 - 2406.
  11. Di Gaspero, L., Di Tollo, G., Roli, A., and Schaerf, A. (2011). Hybrid metaheuristics for constrained portfolio selection problems. Quantitative Finance, 11(10):1473-1487.
  12. Di Tollo, G. and Roli, A. (2008). Metaheuristics for the portfolio selection problem. International Journal of Operationas Research, 5(1):13-35.
  13. Fernandez, A. and Gomez, S. (2007). Portfolio selection using neural networks. Computers & Operations Research, 34(4):1177 - 1191.
  14. Golmakani, H. R. and Fazel, M. (2011). Constrained portfolio selection using particle swarm optimization. Expert Systems with Applications, 38(7):8327 - 8335.
  15. Jansen, R. and van Dijk, R. (2002). Optimal benchmark tracking with small portfolios. The Journal of Portfolio Management, 28(2):33-39.
  16. Jin, Y., Qu, R., and Atkin, J. (2014). A population-based incremental learning method for constrained portfolio optimisation. In Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 2014 16th International Symposium on, pages 212-219.
  17. Jobst, N., Horniman, M., Lucas, C., and Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative Finance, 1(5):489-501.
  18. Kellerer, H., Mansini, R., and Speranza, M. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1- 4):287-304.
  19. Lin, C.-C. and Liu, Y.-T. (2008). Genetic algorithms for portfolio selection problems with minimum transaction lots. European Journal of Operational Research, 185(1):393 - 404.
  20. Lwin, K., Qu, R., and Kendall, G. (2014). A learningguided multi-objective evolutionary algorithm for constrained portfolio optimization. Applied Soft Computing, 24:757 - 772.
  21. Mansini, R. and Speranza, M. G. (1999). Heuristic algorithms for the portfolio selection problem with minimum transaction lots. European Journal of Operational Research, 114(2):219 - 233.
  22. Maringer, D. (2008). Heuristic optimization for portfolio management [application notes]. Computational Intelligence Magazine, IEEE, 3(4):31 -34.
  23. Maringer, D. and Kellerer, H. (2003). Optimization of cardinality constrained portfolios with a hybrid local search algorithm. OR Spectrum, 25(4):481-495.
  24. Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1):pp. 77-91.
  25. Metaxiotis, K. and Liagkouras, K. (2012). Multiobjective evolutionary algorithms for portfolio management: A comprehensive literature review. Expert Systems with Applications, 39(14):11685 - 11698.
  26. Ruiz-Torrubiano, R. and Suarez, A. (2010). Hybrid approaches and dimensionality reduction for portfolio selection with cardinality constraints. Computational Intelligence Magazine, IEEE, 5(2):92-107.
  27. Schaerf, A. (2002). Local search techniques for constrained portfolio selection problems. Computational Economics, 20:177-190.
  28. Skolpadungket, P., Dahal, K., and Harnpornchai, N. (2007). Portfolio optimization using multi-objective genetic algorithms. In Evolutionary Computation, 2007. CEC 2007. IEEE Congress on, pages 516-523.
  29. Speranza, M. G. (1996). A heuristic algorithm for a portfolio optimization model applied to the milan stock market. Computers & Operations Research, 23(5):433 - 441.
  30. Streichert, F., Ulmer, H., and Zell, A. (2004). Evaluating a hybrid encoding and three crossover operators on the constrained portfolio selection problem. In Evolutionary Computation, 2004. CEC2004. Congress on, volume 1, pages 932-939.
  31. Vijayalakshmi Pai, G. and Michel, T. (2009). Evolutionary optimization of constrained k -means clustered assets for diversification in small portfolios. Evolutionary Computation, IEEE Transactions on, 13(5):1030- 1053.
  32. Woodside-Oriakhi, M., Lucas, C., and Beasley, J. (2011). Heuristic algorithms for the cardinality constrained efficient frontier. European Journal of Operational Research, 213(3):538 - 550.
  33. Xu, R.-t., Zhang, J., Liu, O., and Huang, R.-Z. (2010). An estimation of distribution algorithm based portfolio selection approach. In Technologies and Applications of Artificial Intelligence (TAAI), 2010 International Conference on, pages 305-313.

Paper Citation

in Harvard Style

Jin Y., Qu R. and Atkin J. (2016). Constrained Portfolio Optimisation: The State-of-the-Art Markowitz Models . In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES, ISBN 978-989-758-171-7, pages 388-395. DOI: 10.5220/0005758303880395

in Bibtex Style

author={Yan Jin and Rong Qu and Jason Atkin},
title={Constrained Portfolio Optimisation: The State-of-the-Art Markowitz Models},
booktitle={Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

in EndNote Style

JO - Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,
TI - Constrained Portfolio Optimisation: The State-of-the-Art Markowitz Models
SN - 978-989-758-171-7
AU - Jin Y.
AU - Qu R.
AU - Atkin J.
PY - 2016
SP - 388
EP - 395
DO - 10.5220/0005758303880395