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April  2019, 15(2): 909-932. doi: 10.3934/jimo.2018077

A difference of convex optimization algorithm for piecewise linear regression

Adil Bagirov 1,, , Sona Taheri 1, and  Soodabeh Asadi 2,

1. 

Faculty of Science and Technology, Federation University Australia, Victoria, Australia
2. 

Department of Mathematics, Shahrekord University, Iran

* Corresponding author: Adil Bagirov

Received  August 2017 Revised  February 2018 Published  April 2019 Early access  June 2018

Fund Project: The first and second authors are supported by Australian Research Council's Discovery Projects funding scheme (Project No. DP140103213).

The problem of finding a continuous piecewise linear function approximating a regression function is considered. This problem is formulated as a nonconvex nonsmooth optimization problem where the objective function is represented as a difference of convex (DC) functions. Subdifferentials of DC components are computed and an algorithm is designed based on these subdifferentials to find piecewise linear functions. The algorithm is tested using some synthetic and real world data sets and compared with other regression algorithms.

Keywords: Regression analysis, nonsmooth optimization, nonconvex optimization, DC optimization, subdifferential.
Mathematics Subject Classification: 65K05, 90C90.
Citation: Adil Bagirov, Sona Taheri, Soodabeh Asadi. A difference of convex optimization algorithm for piecewise linear regression. Journal of Industrial & Management Optimization, 2019, 15 (2) : 909-932. doi: 10.3934/jimo.2018077
References:
[1]

L. T. H. An and P. D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Annals of Operations Research, 133 (2005), 23-46.  doi: 10.1007/s10479-004-5022-1.  Google Scholar

[2]

K. Bache and M. Lichman, UCI Machine Learning Repository, University of California, Irvine, School of Information and Computer Sciences, http://archive.ics.uci.edu/ml, 2013. Google Scholar

[3]

A. M. Bagirov, C. Clausen and M. Kohler, Estimation of a regression function by maxima of minima of linear functions, IEEE Transactions on Information Theory, 55 (2009), 833-845.  doi: 10.1109/TIT.2008.2009835.  Google Scholar

[4]

A. M. Bagirov, C. Clausen and M. Kohler, An algorithm for the estimation of a regression function by continuous piecewise linear functions, Computational Optimization and Applications, 45 (2010), 159-179.  doi: 10.1007/s10589-008-9174-9.  Google Scholar

[5]

A. M. Bagirov, C. Clausen and M. Kohler, An l2-boosting algorithm for estimation of a regression function, IEEE Transactions on Information Theory, 56 (2010), 1417-1429.  doi: 10.1109/TIT.2009.2039161.  Google Scholar

[6]

A. M. Bagirov, B. Karasözen and M. Sezer, Discrete gradient method: A derivative-free method for nonsmooth optimization, Journal of Optimization Theory and Applications, 137 (2008), 317-334.  doi: 10.1007/s10957-007-9335-5.  Google Scholar

[7]

A. M. Bagirov, N. Karmitsa and M. Mäkelä, Introduction to Nonsmooth Optimization, Cham, Springer, 2014. doi: 10.1007/978-3-319-08114-4.  Google Scholar

[8]

A. M. Bagirov, S. Taheri and J. Ugon, Nonsmooth DC programming approach to the minimum sum-of-squares clustering problems, Pattern Recognition, 53 (2016), 12-24.  doi: 10.1016/j.patcog.2015.11.011.  Google Scholar

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S. G. Bartels, L. Kuntz and S. Scholtes, Continuous selections of linear functions and nonsmooth critical point theory, Nonlinear Analysis: Theory, Methods & Applications, 24 (1995), 385-407.  doi: 10.1016/0362-546X(95)91645-6.  Google Scholar

[10]

L. Breiman, J. H. Friedman, C. J. Stone and R. A. Olshen, Classification and Regression Trees, CRC press, 1984.  Google Scholar

[11]

R. Collobert and S. Bengio, SVMTorch: Support vector machines for large-scale regression problems, Journal of Machine Learning Research, 1 (2001), 143-160.  doi: 10.1162/15324430152733142.  Google Scholar

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V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Springer, 1995.  Google Scholar

[13]

J. H. Fridedman, Multivariate adaptive regression splines (with discussion), Annals of Statistics, 19 (1991), 79-141.  doi: 10.1214/aos/1176347963.  Google Scholar

[14]

J. Friedman, T. Hastie and R. Tibshirani, The Elements of Statistical Learning, Springer, Berlin, 2001. doi: 10.1007/978-0-387-21606-5.  Google Scholar

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V. V. Gorokhovik, O. I. Zorko and G. Birkhoff, Piecewise affine functions and polyhedral sets, Optimization, 31 (1994), 209-221.  doi: 10.1080/02331939408844018.  Google Scholar

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L. Györfi, M. Kohler, A. Krzyźak and H. Walk, A Distribution-Free Theory of Nonparametric Regression, Springer Series in Statistics. Springer, Heldelberg, 2002. doi: 10.1007/b97848.  Google Scholar

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R. Horst and N. V. Thoai, DC programming: Overview, Journal of Optimization Theory and Applications, 103 (1999), 1-43.  doi: 10.1023/A:1021765131316.  Google Scholar

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K. Joki, A. M. Bagirov, N. Karmitsa and M. M. Mäkelä, A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes, Journal of Global Optimization, 68 (2017), 501-535.  doi: 10.1007/s10898-016-0488-3.  Google Scholar

[19]

D. Meyer, E. Dimitriadou, K. Hornik, A. Weingessel, F. Leisch, C. C. Chang and C. C. Lin, Misc functions of the department of statistics, probability theory group, TU Wien, Package "e1071", http://cran.r-project.org/web/packages/e1071/e1071.pdf, 2017. Google Scholar

[20]

S. Milborrow, Multivariate adaptive regression splines, Package "earth", http://cran.r-project.org/web/packages/earth/earth.pdf, 2017. Google Scholar

[21]

J. Nash and J. Sutcliffe, River flow forecasting through conceptual models part I-A discussion of principles, Journal of Hydrology, 10 (1970), 282-290.  doi: 10.1016/0022-1694(70)90255-6.  Google Scholar

[22]

B. Ripley and W. Venables, Feed-forward neural networks and multinomial log-linear models, Package "nnet", http://cran.r-project.org/web/packages/nnet/nnet.pdf, 2016. Google Scholar

[23]

A. J. Smola and B. Schölkopf, A tutorial on support vector regression, Statistics and Computing, 14 (2004), 199-222.  doi: 10.1023/B:STCO.0000035301.49549.88.  Google Scholar

[24]

P. D. Tao and L. T. H. An, Convex analysis approach to DC programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22 (1997), 289-355.   Google Scholar

[25]

H. Tuy, Convex Analysis and Global Optimization, Nonconvex Optimization and Its Applications, Vol. 22. Kluwer, Dordrecht, 1998. doi: 10.1007/978-1-4757-2809-5.  Google Scholar

[26]

P. H. Wolfe, Finding the nearest point in a polytope, Mathematical Programming, 11 (1976), 128-149.  doi: 10.1007/BF01580381.  Google Scholar

show all references

References:
[1]

L. T. H. An and P. D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Annals of Operations Research, 133 (2005), 23-46.  doi: 10.1007/s10479-004-5022-1.  Google Scholar

[2]

K. Bache and M. Lichman, UCI Machine Learning Repository, University of California, Irvine, School of Information and Computer Sciences, http://archive.ics.uci.edu/ml, 2013. Google Scholar

[3]

A. M. Bagirov, C. Clausen and M. Kohler, Estimation of a regression function by maxima of minima of linear functions, IEEE Transactions on Information Theory, 55 (2009), 833-845.  doi: 10.1109/TIT.2008.2009835.  Google Scholar

[4]

A. M. Bagirov, C. Clausen and M. Kohler, An algorithm for the estimation of a regression function by continuous piecewise linear functions, Computational Optimization and Applications, 45 (2010), 159-179.  doi: 10.1007/s10589-008-9174-9.  Google Scholar

[5]

A. M. Bagirov, C. Clausen and M. Kohler, An l2-boosting algorithm for estimation of a regression function, IEEE Transactions on Information Theory, 56 (2010), 1417-1429.  doi: 10.1109/TIT.2009.2039161.  Google Scholar

[6]

A. M. Bagirov, B. Karasözen and M. Sezer, Discrete gradient method: A derivative-free method for nonsmooth optimization, Journal of Optimization Theory and Applications, 137 (2008), 317-334.  doi: 10.1007/s10957-007-9335-5.  Google Scholar

[7]

A. M. Bagirov, N. Karmitsa and M. Mäkelä, Introduction to Nonsmooth Optimization, Cham, Springer, 2014. doi: 10.1007/978-3-319-08114-4.  Google Scholar

[8]

A. M. Bagirov, S. Taheri and J. Ugon, Nonsmooth DC programming approach to the minimum sum-of-squares clustering problems, Pattern Recognition, 53 (2016), 12-24.  doi: 10.1016/j.patcog.2015.11.011.  Google Scholar

[9]

S. G. Bartels, L. Kuntz and S. Scholtes, Continuous selections of linear functions and nonsmooth critical point theory, Nonlinear Analysis: Theory, Methods & Applications, 24 (1995), 385-407.  doi: 10.1016/0362-546X(95)91645-6.  Google Scholar

[10]

L. Breiman, J. H. Friedman, C. J. Stone and R. A. Olshen, Classification and Regression Trees, CRC press, 1984.  Google Scholar

[11]

R. Collobert and S. Bengio, SVMTorch: Support vector machines for large-scale regression problems, Journal of Machine Learning Research, 1 (2001), 143-160.  doi: 10.1162/15324430152733142.  Google Scholar

[12]

V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Springer, 1995.  Google Scholar

[13]

J. H. Fridedman, Multivariate adaptive regression splines (with discussion), Annals of Statistics, 19 (1991), 79-141.  doi: 10.1214/aos/1176347963.  Google Scholar

[14]

J. Friedman, T. Hastie and R. Tibshirani, The Elements of Statistical Learning, Springer, Berlin, 2001. doi: 10.1007/978-0-387-21606-5.  Google Scholar

[15]

V. V. Gorokhovik, O. I. Zorko and G. Birkhoff, Piecewise affine functions and polyhedral sets, Optimization, 31 (1994), 209-221.  doi: 10.1080/02331939408844018.  Google Scholar

[16]

L. Györfi, M. Kohler, A. Krzyźak and H. Walk, A Distribution-Free Theory of Nonparametric Regression, Springer Series in Statistics. Springer, Heldelberg, 2002. doi: 10.1007/b97848.  Google Scholar

[17]

R. Horst and N. V. Thoai, DC programming: Overview, Journal of Optimization Theory and Applications, 103 (1999), 1-43.  doi: 10.1023/A:1021765131316.  Google Scholar

[18]

K. Joki, A. M. Bagirov, N. Karmitsa and M. M. Mäkelä, A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes, Journal of Global Optimization, 68 (2017), 501-535.  doi: 10.1007/s10898-016-0488-3.  Google Scholar

[19]

D. Meyer, E. Dimitriadou, K. Hornik, A. Weingessel, F. Leisch, C. C. Chang and C. C. Lin, Misc functions of the department of statistics, probability theory group, TU Wien, Package "e1071", http://cran.r-project.org/web/packages/e1071/e1071.pdf, 2017. Google Scholar

[20]

S. Milborrow, Multivariate adaptive regression splines, Package "earth", http://cran.r-project.org/web/packages/earth/earth.pdf, 2017. Google Scholar

[21]

J. Nash and J. Sutcliffe, River flow forecasting through conceptual models part I-A discussion of principles, Journal of Hydrology, 10 (1970), 282-290.  doi: 10.1016/0022-1694(70)90255-6.  Google Scholar

[22]

B. Ripley and W. Venables, Feed-forward neural networks and multinomial log-linear models, Package "nnet", http://cran.r-project.org/web/packages/nnet/nnet.pdf, 2016. Google Scholar

[23]

A. J. Smola and B. Schölkopf, A tutorial on support vector regression, Statistics and Computing, 14 (2004), 199-222.  doi: 10.1023/B:STCO.0000035301.49549.88.  Google Scholar

[24]

P. D. Tao and L. T. H. An, Convex analysis approach to DC programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22 (1997), 289-355.   Google Scholar

[25]

H. Tuy, Convex Analysis and Global Optimization, Nonconvex Optimization and Its Applications, Vol. 22. Kluwer, Dordrecht, 1998. doi: 10.1007/978-1-4757-2809-5.  Google Scholar

[26]

P. H. Wolfe, Finding the nearest point in a polytope, Mathematical Programming, 11 (1976), 128-149.  doi: 10.1007/BF01580381.  Google Scholar

Figure 1.  Illustration of the PWLREG algorithm: (a) (1, 1); (b) (2, 1); (c) (2, 2); (d) (3, 2); (e) (4, 2); (f) (5, 2)
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Figure 2.  1 input variable and 500 instances; (a) PWLREG; (b) SVM(Lin); (c) SVM(RBF)
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Figure 3.  1 input variable and 5000 instances; (a) PWLREG; (b) SVM(Lin); (c) SVM(RBF)
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Figure 4.  (a) 2 input variables and 5000 instances (a) PWLREG; (b) SVM(Lin); (c) SVM(RBF)
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Figure 5.  (a) 2 input variables and 5000 instances (a) PWLREG; (b) SVM(Lin); (c) SVM(RBF)
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Figure 6.  (a) CPU time vs the number of input variables; (b) CPU time vs the number of instances
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Table 1.  RMSE results and CPU time for PWLREG on synthetic data sets
$m$ $n$ Train Test CPU   $m$ $n$ Train Test CPU
$5\cdot 10^2$ 1 1.089 1.084 0.09   $3\cdot 10^3$ 1 1.138 1.113 0.37
$5\cdot 10^2$ 2 0.890 0.856 0.14   $3\cdot 10^3$ 2 0.596 0.587 5.90
$5\cdot 10^2$ 3 0.982 1.146 0.87   $3\cdot 10^3$ 3 1.063 1.101 3.37
$5\cdot 10^2$ 4 1.143 1.395 0.59   $3\cdot 10^3$ 4 1.168 1.289 4.32
$5\cdot 10^2$ 7 1.307 1.409 0.39   $3\cdot 10^3$ 7 1.310 1.353 2.12
$5\cdot 10^2$ 10 1.311 1.406 0.28   $3\cdot 10^3$ 10 1.314 1.355 1.64
$1\cdot 10^3$ 1 1.061 1.101 0.14   $5\cdot 10^3$ 1 1.120 1.118 0.62
$1\cdot 10^3$ 2 0.665 0.725 1.70   $5\cdot 10^3$ 2 0.706 0.692 9.10
$1\cdot 10^3$ 3 1.003 1.077 1.22   $5\cdot 10^3$ 3 1.067 1.038 6.96
$1\cdot 10^3$ 4 1.312 1.225 0.59   $5\cdot 10^3$ 4 1.263 1.271 7.07
$1\cdot 10^3$ 7 1.368 1.257 0.59   $5\cdot 10^3$ 7 1.337 1.323 2.56
$1\cdot 10^3$ 10 1.363 1.263 0.92   $5\cdot 10^3$ 10 1.337 1.322 4.10
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$m$ $n$ Train Test CPU   $m$ $n$ Train Test CPU
$5\cdot 10^2$ 1 1.089 1.084 0.09   $3\cdot 10^3$ 1 1.138 1.113 0.37
$5\cdot 10^2$ 2 0.890 0.856 0.14   $3\cdot 10^3$ 2 0.596 0.587 5.90
$5\cdot 10^2$ 3 0.982 1.146 0.87   $3\cdot 10^3$ 3 1.063 1.101 3.37
$5\cdot 10^2$ 4 1.143 1.395 0.59   $3\cdot 10^3$ 4 1.168 1.289 4.32
$5\cdot 10^2$ 7 1.307 1.409 0.39   $3\cdot 10^3$ 7 1.310 1.353 2.12
$5\cdot 10^2$ 10 1.311 1.406 0.28   $3\cdot 10^3$ 10 1.314 1.355 1.64
$1\cdot 10^3$ 1 1.061 1.101 0.14   $5\cdot 10^3$ 1 1.120 1.118 0.62
$1\cdot 10^3$ 2 0.665 0.725 1.70   $5\cdot 10^3$ 2 0.706 0.692 9.10
$1\cdot 10^3$ 3 1.003 1.077 1.22   $5\cdot 10^3$ 3 1.067 1.038 6.96
$1\cdot 10^3$ 4 1.312 1.225 0.59   $5\cdot 10^3$ 4 1.263 1.271 7.07
$1\cdot 10^3$ 7 1.368 1.257 0.59   $5\cdot 10^3$ 7 1.337 1.323 2.56
$1\cdot 10^3$ 10 1.363 1.263 0.92   $5\cdot 10^3$ 10 1.337 1.322 4.10
Table 2.  Results for PWLREG algorithm with the different levels of noise
(1, 1) (2, 1) (2, 2) (3, 2) (4, 2) (5, 2)
$\sigma=0$
RMSE 1.094 1.094 0.063 0.058 0.055 0.055
MAE 0.842 0.842 0.016 0.016 0.015 0.015
MSE 1.198 1.198 0.004 0.003 0.003 0.003
CE 0.013 0.013 0.997 0.997 0.998 0.998
r 0.114 0.114 0.998 0.999 0.999 0.999
$\sigma=0.5$
RMSE 1.221 1.221 0.503 0.503 0.503 0.503
MAE 0.937 0.937 0.404 0.404 0.404 0.404
MSE 1.491 1.492 0.254 0.254 0.254 0.254
CE 0.011 0.011 0.832 0.832 0.832 0.832
r 0.105 0.105 0.912 0.912 0.912 0.912
$\sigma=1$
RMSE 1.498 1.498 1.242 1.000 1.000 1.000
MAE 1.169 1.169 0.993 0.801 0.801 0.801
MSE 2.246 2.247 1.546 1.003 1.004 1.005
CE 0.008 0.008 0.318 0.558 0.558 0.558
r 0.087 0.087 0.564 0.747 0.747 0.747
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(1, 1) (2, 1) (2, 2) (3, 2) (4, 2) (5, 2)
$\sigma=0$
RMSE 1.094 1.094 0.063 0.058 0.055 0.055
MAE 0.842 0.842 0.016 0.016 0.015 0.015
MSE 1.198 1.198 0.004 0.003 0.003 0.003
CE 0.013 0.013 0.997 0.997 0.998 0.998
r 0.114 0.114 0.998 0.999 0.999 0.999
$\sigma=0.5$
RMSE 1.221 1.221 0.503 0.503 0.503 0.503
MAE 0.937 0.937 0.404 0.404 0.404 0.404
MSE 1.491 1.492 0.254 0.254 0.254 0.254
CE 0.011 0.011 0.832 0.832 0.832 0.832
r 0.105 0.105 0.912 0.912 0.912 0.912
$\sigma=1$
RMSE 1.498 1.498 1.242 1.000 1.000 1.000
MAE 1.169 1.169 0.993 0.801 0.801 0.801
MSE 2.246 2.247 1.546 1.003 1.004 1.005
CE 0.008 0.008 0.318 0.558 0.558 0.558
r 0.087 0.087 0.564 0.747 0.747 0.747
Table 3.  The brief description of real world data sets
Data set $m$ $n$
1. Airfoil Self-noise 1503 5
2. Red Wine Quality 1599 11
3. White Wine Quality 4898 11
4. Combined Cycle Power Plant 9568 4
5. Physicochemical Properties 45730 9
 of Protein Tertiary Structure
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Data set $m$ $n$
1. Airfoil Self-noise 1503 5
2. Red Wine Quality 1599 11
3. White Wine Quality 4898 11
4. Combined Cycle Power Plant 9568 4
5. Physicochemical Properties 45730 9
 of Protein Tertiary Structure
Table 4.  Prediction performances of algorithms for real world data sets
PWLREG MLR SVM(Lin) SVM(RBF) MARS ANN
Airfoil Self-noise
RMSE 4.445 5.024 5.037 4.710 5.672 5.499
MAE 3.404 3.905 3.877 3.327 4.614 4.342
MSE 24.641 25.750 25.885 22.941 32.822 31.059
CE 0.653 0.558 0.556 0.611 0.437 0.471
r 0.875 0.761 0.760 0.825 0.704 0.709
Red Wine Quality
RMSE 0.639 0.643 0.657 0.735 0.658 0.636
MAE 0.489 0.490 0.489 0.566 0.503 0.482
MSE 0.480 0.430 0.450 0.569 0.450 0.423
CE 0.319 0.312 0.282 0.101 0.280 0.326
r 0.582 0.570 0.537 0.394 0.555 0.585
White Wine Quality
RMSE 0.637 0.690 0.717 0.675 0.713 0.693
MAE 0.508 0.532 0.551 0.522 0.527 0.545
MSE 0.449 0.482 0.521 0.463 0.515 0.487
CE 0.330 0.216 0.153 0.249 0.164 0.209
r 0.616 0.494 0.467 0.521 0.537 0.520
Combined Cycle Power Plant
RMSE 4.163 4.484 4.503 3.907 4.187 4.283
MAE 3.276 3.572 3.559 2.920 3.295 3.396
MSE 17.702 20.159 20.341 15.377 17.577 18.411
CE 0.942 0.933 0.932 0.949 0.942 0.939
r 0.970 0.966 0.966 0.975 0.971 0.969
Physicochemical Properties Protein
RMSE 4.802 5.203 5.302 4.321 5.044 5.783
MAE 3.840 4.374 4.240 3.018 4.146 4.987
MSE 23.061 27.101 28.145 18.699 25.470 33.487
CE 0.390 0.285 0.257 0.507 0.328 0.116
r 0.625 0.534 0.528 0.719 0.573 0.341
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PWLREG MLR SVM(Lin) SVM(RBF) MARS ANN
Airfoil Self-noise
RMSE 4.445 5.024 5.037 4.710 5.672 5.499
MAE 3.404 3.905 3.877 3.327 4.614 4.342
MSE 24.641 25.750 25.885 22.941 32.822 31.059
CE 0.653 0.558 0.556 0.611 0.437 0.471
r 0.875 0.761 0.760 0.825 0.704 0.709
Red Wine Quality
RMSE 0.639 0.643 0.657 0.735 0.658 0.636
MAE 0.489 0.490 0.489 0.566 0.503 0.482
MSE 0.480 0.430 0.450 0.569 0.450 0.423
CE 0.319 0.312 0.282 0.101 0.280 0.326
r 0.582 0.570 0.537 0.394 0.555 0.585
White Wine Quality
RMSE 0.637 0.690 0.717 0.675 0.713 0.693
MAE 0.508 0.532 0.551 0.522 0.527 0.545
MSE 0.449 0.482 0.521 0.463 0.515 0.487
CE 0.330 0.216 0.153 0.249 0.164 0.209
r 0.616 0.494 0.467 0.521 0.537 0.520
Combined Cycle Power Plant
RMSE 4.163 4.484 4.503 3.907 4.187 4.283
MAE 3.276 3.572 3.559 2.920 3.295 3.396
MSE 17.702 20.159 20.341 15.377 17.577 18.411
CE 0.942 0.933 0.932 0.949 0.942 0.939
r 0.970 0.966 0.966 0.975 0.971 0.969
Physicochemical Properties Protein
RMSE 4.802 5.203 5.302 4.321 5.044 5.783
MAE 3.840 4.374 4.240 3.018 4.146 4.987
MSE 23.061 27.101 28.145 18.699 25.470 33.487
CE 0.390 0.285 0.257 0.507 0.328 0.116
r 0.625 0.534 0.528 0.719 0.573 0.341
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