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March  2020, 16(2): 991-1008. doi: 10.3934/jimo.2018189

## A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems

 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

Received  May 2018 Revised  August 2018 Published  December 2018

Fund Project: This author's research is supported in part by the Singapore-ETH Centre(SEC), which was established as a collaboration between ETH Zurich and National Research Foundation(NRF) Singapore (FI 370074011) under the auspices of the NRFs Campus for Research Excellence andTechnological Enterprise (CREATE) programme

It is commonly accepted that the estimation error of asset returns' sample mean is much larger than that of sample covariance. In order to hedge the risk raised by the estimation error of the sample mean, we propose a sparse and robust multi-period mean-variance portfolio selection model and show how this proposed model can be equivalently reformulated as a multi-block nonsmooth convex optimization problem. In order to get an optimal strategy, a symmetric Gauss-Seidel based method is implemented. Moreover, we show that the algorithm is globally linearly convergent. The effectiveness of our portfolio selection model and the efficiency of its solution method are demonstrated by empirical experiments on both the synthetic and real datasets.

Citation: Ning Zhang. A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems. Journal of Industrial & Management Optimization, 2020, 16 (2) : 991-1008. doi: 10.3934/jimo.2018189
##### References:
 [1] F. J. Aragón Artacho and M. H. Geoffroy, Characterization of metric regularity of subdifferentials, Journal of Convex Analysis, 15 (2008), 365-380.   Google Scholar [2] A. Ben-Tal, T. Margalit and A. Nemirovski, Robust modeling of multi-stage portfolio problems, High Performance Optimization, 33 (2000), 303-328.  doi: 10.1007/978-1-4757-3216-0_12.  Google Scholar [3] D. Bertsimas and M. Sim, Tractable approximations to robust conic optimization problems, Mathematical Programming, 107 (2006), 5-36.  doi: 10.1007/s10107-005-0677-1.  Google Scholar [4] J. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples, Springer, New York, 2006. doi: 10.1007/978-0-387-31256-9.  Google Scholar [5] G. C. Calafiore, Multi-period portfolio optimization with linear control policies, Automatica, 44 (2008), 2463-2473.  doi: 10.1016/j.automatica.2008.02.007.  Google Scholar [6] L. K. Chan, J. Karceski and J. Lakonishok, On portfolio optimization: Forecasting covariances and choosing the risk model, The Review of Financial Studies, 12 (1999), 937-974.   Google Scholar [7] C. Chen, B. He, Y. Ye and X. Yuan, The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Mathematical Programming, 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.  Google Scholar [8] L. Chen, D. F. Sun and K.-C. Toh, An efficient inexact symmetric Gauss-Seidel based majorized ADMM for high-dimensional convex composite conic programming, Mathematical Programming, 161 (2017), 237-270.  doi: 10.1007/s10107-016-1007-5.  Google Scholar [9] V. K. Chopra and W. T. Ziemba, The effect of errors in means, variances, and covariances on optimal portfolio choice, The Journal of Portfolio Management, 19 (1993), 6-11.   Google Scholar [10] F. H. Clarke, Optimization and Nonsmooth Analysis, volume 5. SIAM, 1990. doi: 10.1137/1.9781611971309.  Google Scholar [11] L. Condat, Fast projection onto the simplex and the $\ell_1$ ball, Mathematical Programming, 158 (2016), 575-585.  doi: 10.1007/s10107-015-0946-6.  Google Scholar [12] X. Cui, J. Gao, X. Li and D. Li, Optimal multi-period mean--variance policy under no-shorting constraint, European Journal of Operational Research, 234 (2014), 459-468.  doi: 10.1016/j.ejor.2013.02.040.  Google Scholar [13] G. B. Dantzig and G. Infanger, Multi-stage stochastic linear programs for portfolio optimization, Annals of Operations Research, 45 (1993), 59-76.  doi: 10.1007/BF02282041.  Google Scholar [14] V. DeMiguel and F. J. Nogales, Portfolio selection with robust estimation, Operations Research, 57 (2009), 560-577.  doi: 10.1287/opre.1080.0566.  Google Scholar [15] A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer Monographs in Mathematics. Springer, Dordrecht, 2009. doi: 10.1007/978-0-387-87821-8.  Google Scholar [16] M. Fazel, T. K. Pong, D. F. Sun and P. Tseng, Hankel matrix rank minimization with applications to system identification and realization, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 946-977.  doi: 10.1137/110853996.  Google Scholar [17] J. Gao and D. Li, Optimal cardinality constrained portfolio selection, Operations Research, 61 (2013), 745-761.  doi: 10.1287/opre.2013.1170.  Google Scholar [18] N. Gülpinar and B. Rustem, Worst-case robust decisions for multi-period mean--variance portfolio optimization, European Journal of Operational Research, 183 (2007), 981-1000.  doi: 10.1016/j.ejor.2006.02.046.  Google Scholar [19] W. W. Hager and H. Zhang, Projection onto a polyhedron that exploits sparsity, SIAM Journal on Optimization, 26 (2016), 1773-1798.  doi: 10.1137/15M102825X.  Google Scholar [20] D. Han, D. F. Sun and L. Zhang, Linear rate convergence of the alternating direction method of multipliers for convex composite programming, Mathematics of Operations Research, 43 (2018), 622-637.  doi: 10.1287/moor.2017.0875.  Google Scholar [21] R. Jagannathan and T. Ma, Risk reduction in large portfolios: Why imposing the wrong constraints helps, The Journal of Finance, 58 (2003), 1651-1683.   Google Scholar [22] J. H. Kim, W. C. Kim and F. J. Fabozzi, Recent developments in robust portfolios with a worst-case approach, Journal of Optimization Theory and Applications, 161 (2014), 103-121.  doi: 10.1007/s10957-013-0329-1.  Google Scholar [23] H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science, 37 (1991), 519-531.   Google Scholar [24] X. Y. Lam, J. Marron, D. Sun and K.-C. Toh, Fast algorithms for large-scale generalized distance weighted discrimination, Journal of Computational and Graphical Statistics, 27 (2018), 368-379.  doi: 10.1080/10618600.2017.1366915.  Google Scholar [25] X. Li, D. F. Sun and K.-C. Toh, A block symmetric {Gauss--Seidel} decomposition theorem for convex composite quadratic programming and its applications, Mathematical Programming, (2018), 1-24, . Google Scholar [26] X. D. Li, D. F. Sun and K.-C. Toh, A schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Mathematical Programming, 155 (2016), 333-373.  doi: 10.1007/s10107-014-0850-5.  Google Scholar [27] X. D. Li, D. F. Sun and K.-C. Toh, QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming, Mathematical Programming Computation, 10 (2018), 703-743.  doi: 10.1007/s12532-018-0137-6.  Google Scholar [28] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.   Google Scholar [29] X. Mei, V. DeMiguel and F. J. Nogales, Multiperiod portfolio optimization with multiple risky assets and general transaction costs, Finance, 69 (2016), 108-120.   Google Scholar [30] R. O. Michaud, The markowitz optimization enigma: Is 'optimized' optimal?, Financial Analysts Journal, 45 (1989), 31-42.   Google Scholar [31] S. M. Robinson, Some continuity properties of polyhedral multifunctions, Mathematical Programming at Oberwolfach, 14 (1981), 206-214.  doi: 10.1007/bfb0120929.  Google Scholar [32] R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research, 1 (1976), 97-116.  doi: 10.1287/moor.1.2.97.  Google Scholar [33] R. T. Rockafellar, Convex Analysis, Princeton university press, 1997.   Google Scholar [34] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of risk, 2 (2000), 21-42.   Google Scholar [35] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, volume 317, Springer Science & Business Media, 1998. Google Scholar [36] M. Sion, On general minimax theorems, Pacific Journal of Mathematics, 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171.  Google Scholar [37] D. F. Sun, K.-C. Toh and L. Q. Yang, A convergent 3-block semiproximal alternating direction method of multipliers for conic programming with 4-type constraints, SIAM Journal on Optimization, 25 (2015), 882-915.  doi: 10.1137/140964357.  Google Scholar [38] Y. Sun, G. Aw, K. L. Teo and G. Zhou, Portfolio optimization using a new probabilistic risk measure, Journal of Industrial and Management Optimization, 11 (2015), 1275-1283.  doi: 10.3934/jimo.2015.11.1275.  Google Scholar [39] R. J.-B. Wets, Stochastic programs with fixed recourse: The equivalent deterministic program, SIAM Review, 16 (1974), 309-339.  doi: 10.1137/1016053.  Google Scholar [40] J. Yang, D. F. Sun and K.-C. Toh, A proximal point algorithm for log-determinant optimization with group lasso regularization, SIAM Journal on Optimization, 23 (2013), 857-893.  doi: 10.1137/120864192.  Google Scholar [41] J. J. Ye and X. Y. Ye, Necessary optimality conditions for optimization problems with variational inequality constraints, Mathematics of Operations Research, 22 (1997), 977-997.  doi: 10.1287/moor.22.4.977.  Google Scholar [42] J. J. Ye and J. Zhang, Enhanced Karush-Kuhn-Tucker condition and weaker constraint qualifications, Mathematical Programming, 139 (2013), 353-381.  doi: 10.1007/s10107-013-0667-7.  Google Scholar [43] J. Zhai and M. Bai, Mean-risk model for uncertain portfolio selection with background risk, Journal of Computational and Applied Mathematics, 330 (2018), 59-69.  doi: 10.1016/j.cam.2017.07.038.  Google Scholar [44] Y. Zhang, X. Li and S. Guo, Portfolio selection problems with markowitz's mean--variance framework: A review of literature, Fuzzy Optimization and Decision Making, 17 (2018), 125-158.  doi: 10.1007/s10700-017-9266-z.  Google Scholar

show all references

##### References:
 [1] F. J. Aragón Artacho and M. H. Geoffroy, Characterization of metric regularity of subdifferentials, Journal of Convex Analysis, 15 (2008), 365-380.   Google Scholar [2] A. Ben-Tal, T. Margalit and A. Nemirovski, Robust modeling of multi-stage portfolio problems, High Performance Optimization, 33 (2000), 303-328.  doi: 10.1007/978-1-4757-3216-0_12.  Google Scholar [3] D. Bertsimas and M. Sim, Tractable approximations to robust conic optimization problems, Mathematical Programming, 107 (2006), 5-36.  doi: 10.1007/s10107-005-0677-1.  Google Scholar [4] J. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples, Springer, New York, 2006. doi: 10.1007/978-0-387-31256-9.  Google Scholar [5] G. C. Calafiore, Multi-period portfolio optimization with linear control policies, Automatica, 44 (2008), 2463-2473.  doi: 10.1016/j.automatica.2008.02.007.  Google Scholar [6] L. K. Chan, J. Karceski and J. Lakonishok, On portfolio optimization: Forecasting covariances and choosing the risk model, The Review of Financial Studies, 12 (1999), 937-974.   Google Scholar [7] C. Chen, B. He, Y. Ye and X. Yuan, The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Mathematical Programming, 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.  Google Scholar [8] L. Chen, D. F. Sun and K.-C. Toh, An efficient inexact symmetric Gauss-Seidel based majorized ADMM for high-dimensional convex composite conic programming, Mathematical Programming, 161 (2017), 237-270.  doi: 10.1007/s10107-016-1007-5.  Google Scholar [9] V. K. Chopra and W. T. Ziemba, The effect of errors in means, variances, and covariances on optimal portfolio choice, The Journal of Portfolio Management, 19 (1993), 6-11.   Google Scholar [10] F. H. Clarke, Optimization and Nonsmooth Analysis, volume 5. SIAM, 1990. doi: 10.1137/1.9781611971309.  Google Scholar [11] L. Condat, Fast projection onto the simplex and the $\ell_1$ ball, Mathematical Programming, 158 (2016), 575-585.  doi: 10.1007/s10107-015-0946-6.  Google Scholar [12] X. Cui, J. Gao, X. Li and D. Li, Optimal multi-period mean--variance policy under no-shorting constraint, European Journal of Operational Research, 234 (2014), 459-468.  doi: 10.1016/j.ejor.2013.02.040.  Google Scholar [13] G. B. Dantzig and G. Infanger, Multi-stage stochastic linear programs for portfolio optimization, Annals of Operations Research, 45 (1993), 59-76.  doi: 10.1007/BF02282041.  Google Scholar [14] V. DeMiguel and F. J. Nogales, Portfolio selection with robust estimation, Operations Research, 57 (2009), 560-577.  doi: 10.1287/opre.1080.0566.  Google Scholar [15] A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer Monographs in Mathematics. Springer, Dordrecht, 2009. doi: 10.1007/978-0-387-87821-8.  Google Scholar [16] M. Fazel, T. K. Pong, D. F. Sun and P. Tseng, Hankel matrix rank minimization with applications to system identification and realization, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 946-977.  doi: 10.1137/110853996.  Google Scholar [17] J. Gao and D. Li, Optimal cardinality constrained portfolio selection, Operations Research, 61 (2013), 745-761.  doi: 10.1287/opre.2013.1170.  Google Scholar [18] N. Gülpinar and B. Rustem, Worst-case robust decisions for multi-period mean--variance portfolio optimization, European Journal of Operational Research, 183 (2007), 981-1000.  doi: 10.1016/j.ejor.2006.02.046.  Google Scholar [19] W. W. Hager and H. Zhang, Projection onto a polyhedron that exploits sparsity, SIAM Journal on Optimization, 26 (2016), 1773-1798.  doi: 10.1137/15M102825X.  Google Scholar [20] D. Han, D. F. Sun and L. Zhang, Linear rate convergence of the alternating direction method of multipliers for convex composite programming, Mathematics of Operations Research, 43 (2018), 622-637.  doi: 10.1287/moor.2017.0875.  Google Scholar [21] R. Jagannathan and T. Ma, Risk reduction in large portfolios: Why imposing the wrong constraints helps, The Journal of Finance, 58 (2003), 1651-1683.   Google Scholar [22] J. H. Kim, W. C. Kim and F. J. Fabozzi, Recent developments in robust portfolios with a worst-case approach, Journal of Optimization Theory and Applications, 161 (2014), 103-121.  doi: 10.1007/s10957-013-0329-1.  Google Scholar [23] H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science, 37 (1991), 519-531.   Google Scholar [24] X. Y. Lam, J. Marron, D. Sun and K.-C. Toh, Fast algorithms for large-scale generalized distance weighted discrimination, Journal of Computational and Graphical Statistics, 27 (2018), 368-379.  doi: 10.1080/10618600.2017.1366915.  Google Scholar [25] X. Li, D. F. Sun and K.-C. Toh, A block symmetric {Gauss--Seidel} decomposition theorem for convex composite quadratic programming and its applications, Mathematical Programming, (2018), 1-24, . Google Scholar [26] X. D. Li, D. F. Sun and K.-C. Toh, A schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Mathematical Programming, 155 (2016), 333-373.  doi: 10.1007/s10107-014-0850-5.  Google Scholar [27] X. D. Li, D. F. Sun and K.-C. Toh, QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming, Mathematical Programming Computation, 10 (2018), 703-743.  doi: 10.1007/s12532-018-0137-6.  Google Scholar [28] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.   Google Scholar [29] X. Mei, V. DeMiguel and F. J. Nogales, Multiperiod portfolio optimization with multiple risky assets and general transaction costs, Finance, 69 (2016), 108-120.   Google Scholar [30] R. O. Michaud, The markowitz optimization enigma: Is 'optimized' optimal?, Financial Analysts Journal, 45 (1989), 31-42.   Google Scholar [31] S. M. Robinson, Some continuity properties of polyhedral multifunctions, Mathematical Programming at Oberwolfach, 14 (1981), 206-214.  doi: 10.1007/bfb0120929.  Google Scholar [32] R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research, 1 (1976), 97-116.  doi: 10.1287/moor.1.2.97.  Google Scholar [33] R. T. Rockafellar, Convex Analysis, Princeton university press, 1997.   Google Scholar [34] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of risk, 2 (2000), 21-42.   Google Scholar [35] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, volume 317, Springer Science & Business Media, 1998. Google Scholar [36] M. Sion, On general minimax theorems, Pacific Journal of Mathematics, 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171.  Google Scholar [37] D. F. Sun, K.-C. Toh and L. Q. Yang, A convergent 3-block semiproximal alternating direction method of multipliers for conic programming with 4-type constraints, SIAM Journal on Optimization, 25 (2015), 882-915.  doi: 10.1137/140964357.  Google Scholar [38] Y. Sun, G. Aw, K. L. Teo and G. Zhou, Portfolio optimization using a new probabilistic risk measure, Journal of Industrial and Management Optimization, 11 (2015), 1275-1283.  doi: 10.3934/jimo.2015.11.1275.  Google Scholar [39] R. J.-B. Wets, Stochastic programs with fixed recourse: The equivalent deterministic program, SIAM Review, 16 (1974), 309-339.  doi: 10.1137/1016053.  Google Scholar [40] J. Yang, D. F. Sun and K.-C. Toh, A proximal point algorithm for log-determinant optimization with group lasso regularization, SIAM Journal on Optimization, 23 (2013), 857-893.  doi: 10.1137/120864192.  Google Scholar [41] J. J. Ye and X. Y. Ye, Necessary optimality conditions for optimization problems with variational inequality constraints, Mathematics of Operations Research, 22 (1997), 977-997.  doi: 10.1287/moor.22.4.977.  Google Scholar [42] J. J. Ye and J. Zhang, Enhanced Karush-Kuhn-Tucker condition and weaker constraint qualifications, Mathematical Programming, 139 (2013), 353-381.  doi: 10.1007/s10107-013-0667-7.  Google Scholar [43] J. Zhai and M. Bai, Mean-risk model for uncertain portfolio selection with background risk, Journal of Computational and Applied Mathematics, 330 (2018), 59-69.  doi: 10.1016/j.cam.2017.07.038.  Google Scholar [44] Y. Zhang, X. Li and S. Guo, Portfolio selection problems with markowitz's mean--variance framework: A review of literature, Fuzzy Optimization and Decision Making, 17 (2018), 125-158.  doi: 10.1007/s10700-017-9266-z.  Google Scholar
Two-period efficient frontiers obtained from robust multi-period mean-variance (RMV) and global minimum-variance (MV) portfolio selection models. $\rho =$1e-5
Comparison between the performance of sGS-sPADMM and direclty extended ADMM on different numbers of periods
Comparison between the performance of sGS-sPADMM and direclty extended ADMM on synthetic data. $n = 1, \alpha = 0.5, \rho = 0.005\lambda_{\max}(C_0)$. All the results are averaged over 10 instances
 $p$ sGS-sPADMM Iter $|$ Time (s) $|$ Res ADMM-d Iter $|$ Time (s) $|$ Res 200 348.8 $|$ 1.6 $|$ 9.95e-06 345.9 $|$ 1.6 $|$ 9.91e-06 300 187.3 $|$ 1.6 $|$ 9.76e-06 325.9 $|$ 2.6 $|$ 9.63e-06 500 175.6 $|$ 4.8 $|$ 9.60e-06 275.3 $|$ 7.2 $|$ 9.45e-06 1000 250.2 $|$ 41.5 $|$ 9.25e-06 296.6 $|$ 48.3 $|$ 9.43e-06 1500 346.7 $|$ 143.2 $|$ 9.66e-06 469.4 $|$ 193.8 $|$ 9.42e-06 2000 406.0 $|$ 300.6 $|$ 4.97e-06 585.2 $|$ 428.0 $|$ 9.47e-06
 $p$ sGS-sPADMM Iter $|$ Time (s) $|$ Res ADMM-d Iter $|$ Time (s) $|$ Res 200 348.8 $|$ 1.6 $|$ 9.95e-06 345.9 $|$ 1.6 $|$ 9.91e-06 300 187.3 $|$ 1.6 $|$ 9.76e-06 325.9 $|$ 2.6 $|$ 9.63e-06 500 175.6 $|$ 4.8 $|$ 9.60e-06 275.3 $|$ 7.2 $|$ 9.45e-06 1000 250.2 $|$ 41.5 $|$ 9.25e-06 296.6 $|$ 48.3 $|$ 9.43e-06 1500 346.7 $|$ 143.2 $|$ 9.66e-06 469.4 $|$ 193.8 $|$ 9.42e-06 2000 406.0 $|$ 300.6 $|$ 4.97e-06 585.2 $|$ 428.0 $|$ 9.47e-06
Comparison between the performance of sGS-sPADMM and direclty extended ADMM on synthetic data. $n = 2, \alpha = 0.5, \rho = 0.005\lambda_{\max}(C_0)$. All the results are averaged over 10 instances
 $p$ sGS-sPADMM Iter $|$ Time (s) $|$ Res ADMM-d Iter $|$ Time (s) $|$ Res 200 386.5 $|$ 2.3 $|$ 9.94e-06 349.4 $|$ 2.0 $|$ 9.93e-06 300 237.4 $|$ 3.2 $|$ 9.86e-06 300.8 $|$ 3.8 $|$ 9.77e-06 500 226.4 $|$ 10.0 $|$ 9.50e-06 330.1 $|$ 13.7 $|$ 9.57e-06 1000 287.0 $|$ 72.3 $|$ 6.76e-06 395.2 $|$ 94.5 $|$ 9.76e-06 1500 387.1 $|$ 245.9 $|$ 6.65e-06 561.9 $|$ 355.2 $|$ 9.55e-06 2000 480.0 $|$ 579.5 $|$ 6.75e-06 596.7 $|$ 693.0 $|$ 9.71e-06
 $p$ sGS-sPADMM Iter $|$ Time (s) $|$ Res ADMM-d Iter $|$ Time (s) $|$ Res 200 386.5 $|$ 2.3 $|$ 9.94e-06 349.4 $|$ 2.0 $|$ 9.93e-06 300 237.4 $|$ 3.2 $|$ 9.86e-06 300.8 $|$ 3.8 $|$ 9.77e-06 500 226.4 $|$ 10.0 $|$ 9.50e-06 330.1 $|$ 13.7 $|$ 9.57e-06 1000 287.0 $|$ 72.3 $|$ 6.76e-06 395.2 $|$ 94.5 $|$ 9.76e-06 1500 387.1 $|$ 245.9 $|$ 6.65e-06 561.9 $|$ 355.2 $|$ 9.55e-06 2000 480.0 $|$ 579.5 $|$ 6.75e-06 596.7 $|$ 693.0 $|$ 9.71e-06
Comparison among the performance of sGS-sPADMM, directly extended ADMM, and sPADMM for solving two-period portfolio selection problem with different $\rho$. {The parameter $\varpi = 5$
 Period $\rho$ nnz sGS-sPADMM Iter $|$ Time (s) $|$ Res ADMM-d Iter $|$ Time (s) $|$ Res sPADMM Iter $|$ Time (s) $|$ Res Jan. 1.0e-06 272 284 $|$ 4.0 $|$ 9.91e-06 283 $|$ 4.1 $|$ 9.98e-06 539 $|$ 31.6 $|$ 1.00e-05 5.0e-06 171 248 $|$ 3.3 $|$ 9.88e-06 373 $|$ 4.5 $|$ 1.00e-05 715 $|$ 42.1 $|$ 9.99e-06 1.0e-05 112 246 $|$ 3.3 $|$ 9.46e-06 500 $|$ 6.2 $|$ 9.98e-06 850 $|$ 52.0 $|$ 9.93e-06 Feb. 1.0e-06 288 249 $|$ 3.4 $|$ 9.69e-06 301 $|$ 3.7 $|$ 9.97e-06 464 $|$ 26.6 $|$ 9.99e-06 5.0e-06 173 250 $|$ 3.3 $|$ 9.48e-06 321 $|$ 4.0 $|$ 9.98e-06 568 $|$ 32.2 $|$ 9.95e-06 1.0e-05 125 258 $|$ 3.6 $|$ 9.01e-06 373 $|$ 4.7 $|$ 9.96e-06 701 $|$ 40.2 $|$ 9.98e-06 Mar. 1.0e-06 269 223 $|$ 3.1 $|$ 9.91e-06 253 $|$ 3.2 $|$ 9.97e-06 484 $|$ 27.8 $|$ 9.88e-06 5.0e-06 153 229 $|$ 3.1 $|$ 9.88e-06 310 $|$ 3.8 $|$ 9.93e-06 596 $|$ 34.0 $|$ 9.98e-06 1.0e-05 122 238 $|$ 3.3 $|$ 9.43e-06 358 $|$ 4.5 $|$ 9.88e-06 594 $|$ 34.1 $|$ 9.96e-06 Apr. 1.0e-06 272 272 $|$ 3.7 $|$ 9.20e-06 316 $|$ 4.0 $|$ 9.98e-06 573 $|$ 33.1 $|$ 9.99e-06 5.0e-06 144 285 $|$ 3.8 $|$ 9.84e-06 477 $|$ 5.8 $|$ 9.94e-06 902 $|$ 51.7 $|$ 9.88e-06 1.0e-05 108 264 $|$ 3.7 $|$ 9.77e-06 548 $|$ 6.9 $|$ 8.67e-06 915 $|$ 52.4 $|$ 9.91e-06 May 1.0e-06 244 290 $|$ 3.9 $|$ 9.80e-06 510 $|$ 6.3 $|$ 9.84e-06 941 $|$ 53.6 $|$ 9.79e-06 5.0e-06 127 337 $|$ 4.5 $|$ 9.75e-06 635 $|$ 7.9 $|$ 9.99e-06 1110 $|$ 63.8 $|$ 9.90e-06 1.0e-05 94 294 $|$ 4.0 $|$ 9.70e-06 765 $|$ 9.6 $|$ 9.98e-06 1354 $|$ 77.2 $|$ 9.98e-06 Jun. 1.0e-06 274 281 $|$ 3.9 $|$ 9.51e-06 448 $|$ 5.6 $|$ 9.73e-06 805 $|$ 45.7 $|$ 9.96e-06 5.0e-06 151 239 $|$ 3.2 $|$ 9.95e-06 478 $|$ 5.9 $|$ 9.95e-06 911 $|$ 51.6 $|$ 9.78e-06 1.0e-05 105 386 $|$ 5.2 $|$ 9.64e-06 509 $|$ 6.4 $|$ 9.97e-06 983 $|$ 55.5 $|$ 9.96e-06
 Period $\rho$ nnz sGS-sPADMM Iter $|$ Time (s) $|$ Res ADMM-d Iter $|$ Time (s) $|$ Res sPADMM Iter $|$ Time (s) $|$ Res Jan. 1.0e-06 272 284 $|$ 4.0 $|$ 9.91e-06 283 $|$ 4.1 $|$ 9.98e-06 539 $|$ 31.6 $|$ 1.00e-05 5.0e-06 171 248 $|$ 3.3 $|$ 9.88e-06 373 $|$ 4.5 $|$ 1.00e-05 715 $|$ 42.1 $|$ 9.99e-06 1.0e-05 112 246 $|$ 3.3 $|$ 9.46e-06 500 $|$ 6.2 $|$ 9.98e-06 850 $|$ 52.0 $|$ 9.93e-06 Feb. 1.0e-06 288 249 $|$ 3.4 $|$ 9.69e-06 301 $|$ 3.7 $|$ 9.97e-06 464 $|$ 26.6 $|$ 9.99e-06 5.0e-06 173 250 $|$ 3.3 $|$ 9.48e-06 321 $|$ 4.0 $|$ 9.98e-06 568 $|$ 32.2 $|$ 9.95e-06 1.0e-05 125 258 $|$ 3.6 $|$ 9.01e-06 373 $|$ 4.7 $|$ 9.96e-06 701 $|$ 40.2 $|$ 9.98e-06 Mar. 1.0e-06 269 223 $|$ 3.1 $|$ 9.91e-06 253 $|$ 3.2 $|$ 9.97e-06 484 $|$ 27.8 $|$ 9.88e-06 5.0e-06 153 229 $|$ 3.1 $|$ 9.88e-06 310 $|$ 3.8 $|$ 9.93e-06 596 $|$ 34.0 $|$ 9.98e-06 1.0e-05 122 238 $|$ 3.3 $|$ 9.43e-06 358 $|$ 4.5 $|$ 9.88e-06 594 $|$ 34.1 $|$ 9.96e-06 Apr. 1.0e-06 272 272 $|$ 3.7 $|$ 9.20e-06 316 $|$ 4.0 $|$ 9.98e-06 573 $|$ 33.1 $|$ 9.99e-06 5.0e-06 144 285 $|$ 3.8 $|$ 9.84e-06 477 $|$ 5.8 $|$ 9.94e-06 902 $|$ 51.7 $|$ 9.88e-06 1.0e-05 108 264 $|$ 3.7 $|$ 9.77e-06 548 $|$ 6.9 $|$ 8.67e-06 915 $|$ 52.4 $|$ 9.91e-06 May 1.0e-06 244 290 $|$ 3.9 $|$ 9.80e-06 510 $|$ 6.3 $|$ 9.84e-06 941 $|$ 53.6 $|$ 9.79e-06 5.0e-06 127 337 $|$ 4.5 $|$ 9.75e-06 635 $|$ 7.9 $|$ 9.99e-06 1110 $|$ 63.8 $|$ 9.90e-06 1.0e-05 94 294 $|$ 4.0 $|$ 9.70e-06 765 $|$ 9.6 $|$ 9.98e-06 1354 $|$ 77.2 $|$ 9.98e-06 Jun. 1.0e-06 274 281 $|$ 3.9 $|$ 9.51e-06 448 $|$ 5.6 $|$ 9.73e-06 805 $|$ 45.7 $|$ 9.96e-06 5.0e-06 151 239 $|$ 3.2 $|$ 9.95e-06 478 $|$ 5.9 $|$ 9.95e-06 911 $|$ 51.6 $|$ 9.78e-06 1.0e-05 105 386 $|$ 5.2 $|$ 9.64e-06 509 $|$ 6.4 $|$ 9.97e-06 983 $|$ 55.5 $|$ 9.96e-06
Comparison among the performance of sGS-sPADMM, directly extended ADMM, and sPADMM for solving three/four-period portfolio selection problem with different $\rho$. Based on 2005 data, the parameter $\varpi = 5$. "nnz" stands for the number of active positions of portfolio
 $n$ $\rho$ nnz sGS-sPADMM Iter $|$ Time (s) $|$ Res ADMM-d Iter $|$ Time (s) $|$ Res sPADMM Iter $|$ Time (s) $|$ Res 2 1.0e-06 270 473 $|$ 11.5 $|$ 9.49e-06 1189 $|$ 27.6 $|$ 9.93e-06 1296 $|$ 289.1 $|$ 9.93e-06 5.0e-06 128 636 $|$ 14.8 $|$ 9.86e-06 1025 $|$ 21.9 $|$ 9.57e-06 1260 $|$ 280.9 $|$ 9.98e-06 1.0e-05 109 585 $|$ 14.2 $|$ 9.99e-06 1356 $|$ 30.4 $|$ 9.29e-06 1234 $|$ 279.5 $|$ 1.00e-05 3 1.0e-06 254 623 $|$ 20.6 $|$ 9.13e-06 1327 $|$ 40.4 $|$ 9.74e-06 1362 $|$ 719.5 $|$ 1.00e-05 5.0e-06 125 939 $|$ 31.2 $|$ 9.68e-06 1522 $|$ 45.5 $|$ 9.57e-06 1346 $|$ 704.4 $|$ 9.93e-06 1.0e-05 103 675 $|$ 23.6 $|$ 9.49e-06 1484 $|$ 42.5 $|$ 1.00e-05 1319 $|$ 694.8 $|$ 9.97e-06
 $n$ $\rho$ nnz sGS-sPADMM Iter $|$ Time (s) $|$ Res ADMM-d Iter $|$ Time (s) $|$ Res sPADMM Iter $|$ Time (s) $|$ Res 2 1.0e-06 270 473 $|$ 11.5 $|$ 9.49e-06 1189 $|$ 27.6 $|$ 9.93e-06 1296 $|$ 289.1 $|$ 9.93e-06 5.0e-06 128 636 $|$ 14.8 $|$ 9.86e-06 1025 $|$ 21.9 $|$ 9.57e-06 1260 $|$ 280.9 $|$ 9.98e-06 1.0e-05 109 585 $|$ 14.2 $|$ 9.99e-06 1356 $|$ 30.4 $|$ 9.29e-06 1234 $|$ 279.5 $|$ 1.00e-05 3 1.0e-06 254 623 $|$ 20.6 $|$ 9.13e-06 1327 $|$ 40.4 $|$ 9.74e-06 1362 $|$ 719.5 $|$ 1.00e-05 5.0e-06 125 939 $|$ 31.2 $|$ 9.68e-06 1522 $|$ 45.5 $|$ 9.57e-06 1346 $|$ 704.4 $|$ 9.93e-06 1.0e-05 103 675 $|$ 23.6 $|$ 9.49e-06 1484 $|$ 42.5 $|$ 1.00e-05 1319 $|$ 694.8 $|$ 9.97e-06
Comparison among the performance of sGS-sPADMM, directly extended ADMM, and sPADMM for solving three/four-period portfolio selection problem with different $\rho$. Based on 2006 data, parameter $\varpi = 5$. "nnz" stands for the number of active positions of portfolio
 $n$ $\rho$ nnz sGS-sPADMM Iter $|$ Time (s) $|$ Res ADMM-d Iter $|$ Time (s) $|$ Res sPADMM Iter $|$ Time (s) $|$ Res 2 1.0e-06 237 490 $|$ 11.3 $|$ 9.57e-06 1146 $|$ 25.5 $|$ 9.89e-06 1924 $|$ 449.4 $|$ 9.83e-06 5.0e-06 138 681 $|$ 14.8 $|$ 9.82e-06 1297 $|$ 26.0 $|$ 9.99e-06 1584 $|$ 338.2 $|$ 1.00e-05 1.0e-05 120 596 $|$ 13.2 $|$ 9.86e-06 1213 $|$ 24.5 $|$ 9.99e-06 1569 $|$ 331.5 $|$ 9.92e-06 3 1.0e-06 225 752 $|$ 22.7 $|$ 8.97e-06 810 $|$ 23.2 $|$ 9.94e-06 1895 $|$ 979.8 $|$ 1.00e-05 5.0e-06 133 809 $|$ 25.5 $|$ 9.74e-06 1057 $|$ 29.0 $|$ 9.97e-06 1828 $|$ 925.8 $|$ 9.89e-06 1.0e-05 120 648 $|$ 23.6 $|$ 9.75e-06 1377 $|$ 39.5 $|$ 9.99e-06 2292 $|$ 1167.0 $|$ 1.00e-05
 $n$ $\rho$ nnz sGS-sPADMM Iter $|$ Time (s) $|$ Res ADMM-d Iter $|$ Time (s) $|$ Res sPADMM Iter $|$ Time (s) $|$ Res 2 1.0e-06 237 490 $|$ 11.3 $|$ 9.57e-06 1146 $|$ 25.5 $|$ 9.89e-06 1924 $|$ 449.4 $|$ 9.83e-06 5.0e-06 138 681 $|$ 14.8 $|$ 9.82e-06 1297 $|$ 26.0 $|$ 9.99e-06 1584 $|$ 338.2 $|$ 1.00e-05 1.0e-05 120 596 $|$ 13.2 $|$ 9.86e-06 1213 $|$ 24.5 $|$ 9.99e-06 1569 $|$ 331.5 $|$ 9.92e-06 3 1.0e-06 225 752 $|$ 22.7 $|$ 8.97e-06 810 $|$ 23.2 $|$ 9.94e-06 1895 $|$ 979.8 $|$ 1.00e-05 5.0e-06 133 809 $|$ 25.5 $|$ 9.74e-06 1057 $|$ 29.0 $|$ 9.97e-06 1828 $|$ 925.8 $|$ 9.89e-06 1.0e-05 120 648 $|$ 23.6 $|$ 9.75e-06 1377 $|$ 39.5 $|$ 9.99e-06 2292 $|$ 1167.0 $|$ 1.00e-05
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