American Institute of Mathematical Sciences

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July  2016, 12(3): 1041-1056. doi: 10.3934/jimo.2016.12.1041

Cardinality constrained portfolio selection problem: A completely positive programming approach

Ye Tian 1, , Shucherng Fang 2, , Zhibin Deng 3, and  Qingwei Jin 4,

1. 

School of Business Administration, Southwestern University of Finance and Economics, Chengdu, 611130
2. 

Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27606, United States
3. 

School of Management, University of Chinese Academy of Sciences, Beijing, 100190
4. 

Department of Management Science and Engineering, Zhejiang University, Hangzhou, Zhejiang 310058

Received  March 2014 Revised  May 2015 Published  September 2015

In this paper, we propose a completely positive programming reformulation of the cardinality constrained portfolio selection problem. By constructing a sequence of computable cones of nonnegative quadratic forms over a union of second-order cones, an $\epsilon$-optimal solution of the original problem can be found in finite iterations using semidefinite programming techniques. In order to obtain a good lower bound efficiently, an adaptive scheme is adopted in our approximation algorithm. The numerical results show that the proposed algorithm can find better approximate and feasible solutions than other known methods in the literature.
Keywords: Cardinality constrained portfolio selection problem, adaptive approximation., completely positive programming, second-order cone.
Mathematics Subject Classification: Primary: 90C26, 90C59, 90C22; Secondary: 30E1.
Citation: Ye Tian, Shucherng Fang, Zhibin Deng, Qingwei Jin. Cardinality constrained portfolio selection problem: A completely positive programming approach. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1041-1056. doi: 10.3934/jimo.2016.12.1041
References:
[1]

D. Bertsimas and R. Shioda, Algorithm for cardinality-constrained quadratic optimization, Computational Optimization and Applications, 43 (2009), 1-22. doi: 10.1007/s10589-007-9126-9.

[2]

D. Bienstock, Computational study of a family of mixed-integer quadratic programming problems, Mathematical Programming, 74 (1996), 121-140. doi: 10.1007/BF02592208.

[3]

P. Bonami and M. Lejeune, An exact solution approach for portfolio optimization problems under stochastic and integer constraints, Operations Research, 57 (2009), 650-670. doi: 10.1287/opre.1080.0599.

[4]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441.

[5]

S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z.

[6]

T. Chang, N. Meade, J. Beasley and Y. Sharaiha, Heuristics for cardinality constrained portfolio optimization, Computational Operations Research, 27 (2000), 1271-1302.

[7]

X. Cui, X. Sun and D. Sha, An empirical study on discrete optimization models for portfolio selection, Journal of Industrial and Managment Optimization, 5 (2009), 33-46. doi: 10.3934/jimo.2009.5.33.

[8]

X. Cui, X. Zheng, S. Zhu and X. Sun, Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio slection problems, Journal of Global Optimization, 56 (2013), 1409-1423. doi: 10.1007/s10898-012-9842-2.

[9]

Z. Deng, S.-C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme, European Journal of Operational Research, 229 (2013), 21-28. doi: 10.1016/j.ejor.2013.02.031.

[10]

E. Elton and M. Gruber, Modern Portfolio Theory and Investment Analysis, 2nd edition, John Wiley & Sons, Inc, Hoboken, 1984.

[11]

A. Frangioni and C. Gentile, Perspective cuts for a class of convex 0-1 mixed integer programs, Mathematical Programming, 106 (2006), 225-236. doi: 10.1007/s10107-005-0594-3.

[12]

A. Frangioni and C. Gentile, SDP diagonalizations and perspective cuts for a class of nonseparable MIQP, Operations Research Letters, 35 (2007), 181-185. doi: 10.1016/j.orl.2006.03.008.

[13]

A. Frangioni and C. Gentile, A computational comparison of reformulations of the perspective relaxation: SOCP vs. cutting planes, Operations Research Letters, 37 (2009), 206-210. doi: 10.1016/j.orl.2009.02.003.

[14]

J. Gao and D. Li, Optimal cardinality constrained portfolio selecton, Operations Research, 61 (2013), 745-761. doi: 10.1287/opre.2013.1170.

[15]

M. Grant and S. Boyd, CVX: matlab software for disciplined programming, version 1.2, http://cvxr.com/cvx, 2010.

[16]

Q. Jin, Y. Tian, Z. Deng, S.-C. Fang and W. Xing, Exact computable representation of some second-order cone constrained quadratic programming problems, Journal of the Operations Research Society of China, 1 (2013), 107-134. doi: 10.1007/s40305-013-0009-8.

[17]

N. Jobst, M. Horniman, C. Luca and G. Mitra, Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints, Quantitative Finance, 1 (2001), 489-501. doi: 10.1088/1469-7688/1/5/301.

[18]

H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science, 37 (1991), 519-531. doi: 10.1287/mnsc.37.5.519.

[19]

P. Leung, H. Ng and W. Wong, An improved estimation to make Markowitz's portfolio optimization theory users friendly and estimation accurate with application on the US stock market investment, European Journal of Operational Research, 222 (2012), 85-95. doi: 10.1016/j.ejor.2012.04.003.

[20]

P. Lin, Portfolo optimization and risk measurement based on non-dominated sorting genetic algorithm, Journal of Industrial and Management Optimization, 8 (2012), 549-564. doi: 10.3934/jimo.2012.8.549.

[21]

D. Maringer and H. Kellerer, Optimization of cardinality constrained portfolios with a hybrid local search algorithm, OR Spectrum, 25 (2003), 481-495. doi: 10.1007/s00291-003-0139-1.

[22]

H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.

[23]

G. Mitra, F. Ellison and A. Scowcroft, Quadratic programming for portfolio planning: Insights into algorithmic and computational issues. Part II: Processing of portfolio planning models with discrete constraints, Journal of Asset Management, 8 (2007), 249-258. doi: 10.1057/palgrave.jam.2250079.

[24]

K. Murty and S. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Mathematical Programming, 39 (1987), 117-129. doi: 10.1007/BF02592948.

[25]

P. Pardalos and G. Rodgers, Computing aspects of a branch and bound algorithm for quadratic zero-one programming, Computing, 45 (1990), 131-144. doi: 10.1007/BF02247879.

[26]

P. Pardalos and M. Sandström and C. Zopounidis, On the use of optimization models for portfolio selection: A review and some computational results, Computational Economics, 7 (1994), 227-244. doi: 10.1007/BF01299454.

[27]

R. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.

[28]

D. Shaw, S. Liu and L. Kopman, Lagrangian relaxation procedure for cardinality-constrained portfolio optimization, Optimization Methods and Software, 23 (2008), 411-420. doi: 10.1080/10556780701722542.

[29]

J. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267. doi: 10.1287/moor.28.2.246.14485.

[30]

X. Sun, X. Zheng and D. Li, Recent advances in mathematical programming with semi-continuous variables and cardinality constraint, Journal of the Operations Research Society of China, 1 (2013), 55-77. doi: 10.1007/s40305-013-0004-0.

[31]

Y. Tian, S.-C. Fang, Z. Deng and W. Xing, Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming, Journal of Industrial and Management Optimization, 9 (2013), 703-721. doi: 10.3934/jimo.2013.9.703.

[32]

J. Xie, S. He and S. Zhang, Randomized portfolio selection with constraints, Pacific Journal of Optimization, 4 (2008), 87-112.

[33]

Y. Ye and S. Zhang, New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267. doi: 10.1137/S105262340139001X.

[34]

M. Young, A minimax portfolio selction rule with linear programming solution, Management Science, 44 (1998), 673-683.

[35]

Y. Zeng, Z. Li and J. Liu, Optimal strategies of benchmark and mean-variance portfolo selection problems for insurers, Journal of Industral and Management Optimization, 6 (2010), 483-496. doi: 10.3934/jimo.2010.6.483.

[36]

X. Zheng, X. Sun and D. Li, Improving the performance of MIQP solvers for quadratic programs with cardinality and minimum threshold constraints: A semidefinite program approach, INFORMS Journal on Computing, 26 (2014), 690-703. doi: 10.1287/ijoc.2014.0592.

[37]

S. Zymler, B. Rustem and D. Kuhn, Robust portfolio optimization with derivative insurance guarantees, European Journal of Operational Research, 210 (2011), 410-424. doi: 10.1016/j.ejor.2010.09.027.

show all references

References:
[1]

D. Bertsimas and R. Shioda, Algorithm for cardinality-constrained quadratic optimization, Computational Optimization and Applications, 43 (2009), 1-22. doi: 10.1007/s10589-007-9126-9.

[2]

D. Bienstock, Computational study of a family of mixed-integer quadratic programming problems, Mathematical Programming, 74 (1996), 121-140. doi: 10.1007/BF02592208.

[3]

P. Bonami and M. Lejeune, An exact solution approach for portfolio optimization problems under stochastic and integer constraints, Operations Research, 57 (2009), 650-670. doi: 10.1287/opre.1080.0599.

[4]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441.

[5]

S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z.

[6]

T. Chang, N. Meade, J. Beasley and Y. Sharaiha, Heuristics for cardinality constrained portfolio optimization, Computational Operations Research, 27 (2000), 1271-1302.

[7]

X. Cui, X. Sun and D. Sha, An empirical study on discrete optimization models for portfolio selection, Journal of Industrial and Managment Optimization, 5 (2009), 33-46. doi: 10.3934/jimo.2009.5.33.

[8]

X. Cui, X. Zheng, S. Zhu and X. Sun, Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio slection problems, Journal of Global Optimization, 56 (2013), 1409-1423. doi: 10.1007/s10898-012-9842-2.

[9]

Z. Deng, S.-C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme, European Journal of Operational Research, 229 (2013), 21-28. doi: 10.1016/j.ejor.2013.02.031.

[10]

E. Elton and M. Gruber, Modern Portfolio Theory and Investment Analysis, 2nd edition, John Wiley & Sons, Inc, Hoboken, 1984.

[11]

A. Frangioni and C. Gentile, Perspective cuts for a class of convex 0-1 mixed integer programs, Mathematical Programming, 106 (2006), 225-236. doi: 10.1007/s10107-005-0594-3.

[12]

A. Frangioni and C. Gentile, SDP diagonalizations and perspective cuts for a class of nonseparable MIQP, Operations Research Letters, 35 (2007), 181-185. doi: 10.1016/j.orl.2006.03.008.

[13]

A. Frangioni and C. Gentile, A computational comparison of reformulations of the perspective relaxation: SOCP vs. cutting planes, Operations Research Letters, 37 (2009), 206-210. doi: 10.1016/j.orl.2009.02.003.

[14]

J. Gao and D. Li, Optimal cardinality constrained portfolio selecton, Operations Research, 61 (2013), 745-761. doi: 10.1287/opre.2013.1170.

[15]

M. Grant and S. Boyd, CVX: matlab software for disciplined programming, version 1.2, http://cvxr.com/cvx, 2010.

[16]

Q. Jin, Y. Tian, Z. Deng, S.-C. Fang and W. Xing, Exact computable representation of some second-order cone constrained quadratic programming problems, Journal of the Operations Research Society of China, 1 (2013), 107-134. doi: 10.1007/s40305-013-0009-8.

[17]

N. Jobst, M. Horniman, C. Luca and G. Mitra, Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints, Quantitative Finance, 1 (2001), 489-501. doi: 10.1088/1469-7688/1/5/301.

[18]

H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science, 37 (1991), 519-531. doi: 10.1287/mnsc.37.5.519.

[19]

P. Leung, H. Ng and W. Wong, An improved estimation to make Markowitz's portfolio optimization theory users friendly and estimation accurate with application on the US stock market investment, European Journal of Operational Research, 222 (2012), 85-95. doi: 10.1016/j.ejor.2012.04.003.

[20]

P. Lin, Portfolo optimization and risk measurement based on non-dominated sorting genetic algorithm, Journal of Industrial and Management Optimization, 8 (2012), 549-564. doi: 10.3934/jimo.2012.8.549.

[21]

D. Maringer and H. Kellerer, Optimization of cardinality constrained portfolios with a hybrid local search algorithm, OR Spectrum, 25 (2003), 481-495. doi: 10.1007/s00291-003-0139-1.

[22]

H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.

[23]

G. Mitra, F. Ellison and A. Scowcroft, Quadratic programming for portfolio planning: Insights into algorithmic and computational issues. Part II: Processing of portfolio planning models with discrete constraints, Journal of Asset Management, 8 (2007), 249-258. doi: 10.1057/palgrave.jam.2250079.

[24]

K. Murty and S. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Mathematical Programming, 39 (1987), 117-129. doi: 10.1007/BF02592948.

[25]

P. Pardalos and G. Rodgers, Computing aspects of a branch and bound algorithm for quadratic zero-one programming, Computing, 45 (1990), 131-144. doi: 10.1007/BF02247879.

[26]

P. Pardalos and M. Sandström and C. Zopounidis, On the use of optimization models for portfolio selection: A review and some computational results, Computational Economics, 7 (1994), 227-244. doi: 10.1007/BF01299454.

[27]

R. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.

[28]

D. Shaw, S. Liu and L. Kopman, Lagrangian relaxation procedure for cardinality-constrained portfolio optimization, Optimization Methods and Software, 23 (2008), 411-420. doi: 10.1080/10556780701722542.

[29]

J. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267. doi: 10.1287/moor.28.2.246.14485.

[30]

X. Sun, X. Zheng and D. Li, Recent advances in mathematical programming with semi-continuous variables and cardinality constraint, Journal of the Operations Research Society of China, 1 (2013), 55-77. doi: 10.1007/s40305-013-0004-0.

[31]

Y. Tian, S.-C. Fang, Z. Deng and W. Xing, Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming, Journal of Industrial and Management Optimization, 9 (2013), 703-721. doi: 10.3934/jimo.2013.9.703.

[32]

J. Xie, S. He and S. Zhang, Randomized portfolio selection with constraints, Pacific Journal of Optimization, 4 (2008), 87-112.

[33]

Y. Ye and S. Zhang, New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267. doi: 10.1137/S105262340139001X.

[34]

M. Young, A minimax portfolio selction rule with linear programming solution, Management Science, 44 (1998), 673-683.

[35]

Y. Zeng, Z. Li and J. Liu, Optimal strategies of benchmark and mean-variance portfolo selection problems for insurers, Journal of Industral and Management Optimization, 6 (2010), 483-496. doi: 10.3934/jimo.2010.6.483.

[36]

X. Zheng, X. Sun and D. Li, Improving the performance of MIQP solvers for quadratic programs with cardinality and minimum threshold constraints: A semidefinite program approach, INFORMS Journal on Computing, 26 (2014), 690-703. doi: 10.1287/ijoc.2014.0592.

[37]

S. Zymler, B. Rustem and D. Kuhn, Robust portfolio optimization with derivative insurance guarantees, European Journal of Operational Research, 210 (2011), 410-424. doi: 10.1016/j.ejor.2010.09.027.

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