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October  2017, 13(4): 1991-2013. doi: 10.3934/jimo.2017028

Prox-dual regularization algorithm for generalized fractional programs

Mostafa El Haffari 1, and  Ahmed Roubi 1,,

1. 

Laboratoire MISI, Faculté des Sciences et Techniques, Univ. Hassan 1,26000 Settat, Morocco

* Corresponding author:Ahmed Roubi

The authors would like to thank the referees for their valuable comments

Received  September 2015 Revised  February 2017 Published  April 2017

Prox-regularization algorithms for solving generalized fractional programs (GFP) were already considered by several authors. Since the standard dual of a generalized fractional program has not generally the form of GFP, these approaches can not apply directly to the dual problem. In this paper, we propose a primal-dual algorithm for solving convex generalized fractional programs. That is, we use a prox-regularization method to the dual problem that generates a sequence of auxiliary dual problems with unique solutions. So we can avoid the numerical difficulties that can occur if the fractional program does not have a unique solution. Our algorithm is based on Dinkelbach-type algorithms for generalized fractional programming, but uses a regularized parametric auxiliary problem. We establish then the convergence and rate of convergence of this new algorithm.

Keywords: Multi-ratio fractional programs, Dinkelbach-type algorithms, Lagrangian duality, proximal point algorithm.
Mathematics Subject Classification: Primary: 90C32, 49N15, 49M29, 49M37; Secondary: 49K35.
Citation: Mostafa El Haffari, Ahmed Roubi. Prox-dual regularization algorithm for generalized fractional programs. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1991-2013. doi: 10.3934/jimo.2017028
References:
[1]

A. Addou and A. Roubi, Proximal-type methods with generalized bregman functions and applications to generalized fractional programming, Optimization, 59 (2010), 1085-1105.  doi: 10.1080/02331930903395857.

[2]

A. I. Barros, J. B. G. Frenk, S. Schaible and S. Zhang, A new algorithm for generalized fractional programs, Mathematical Programming, 72 (1996), 147-175.  doi: 10.1007/BF02592087.

[3]

A. I. Barros, J. B. G. Frenk, S. Schaible and S. Zhang, Using duality to solve generalized fractional programming problems, Journal of Global Optimization, 8 (1996), 139-170.  doi: 10.1007/BF00138690.

[4]

J. C. Bernard and J. A. Ferland, Convergence of interval-type algorithms for generalized fractional programming, Mathematical Programming, 43 (1989), 349-363.  doi: 10.1007/BF01582298.

[5]

J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming, SIAM Journal on Control and Optimization, 31 (1993), 1340-1359.  doi: 10.1137/0331063.

[6]

J. P. Crouzeix, J. A. Ferland and S. Schaible, Duality in generalized linear fractional programming, Mathematical Programming, 27 (1983), 342-354.  doi: 10.1007/BF02591908.

[7]

J. P. Crouzeix, J. A. Ferland and S. Schaible, An algorithm for generalized fractional programs, Journal of Optimization Theory and Applications, 47 (1985), 35-49.  doi: 10.1007/BF00941314.

[8]

J. P. Crouzeix, J. A. Ferland and S. Schaible, A note on an algorithm for generalized fractional programs, Journal of Optimization Theory and Applications, 50 (1986), 183-187.  doi: 10.1007/BF00938484.

[9]

J. P. Crouzeix and J. A. Ferland, Algorithms for generalized fractional programming, Mathematical Programming, 52 (1991), 191-207.  doi: 10.1007/BF01582887.

[10]

W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498.  doi: 10.1287/mnsc.13.7.492.

[11]

I. Ekeland and R. Temam, Analyse Convexe et Problémes Variationnels, Gauthier-Villars, Paris, Bruxelles, Montréal, 1976.

[12]

J. B. G. Frenk and S. Schaible, Fractional Programming, in ERIM Report Series, (Reference No. ERS-2004-074-LIS) (2004).

[13]

M. Gugat, Prox-regularization methods for generalized fractional programming, Journal of Optimization Theory and Applications, 99 (1998), 691-722.  doi: 10.1023/A:1021759318653.

[14]

O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization, 29 (1991), 403-419. 

[15]

J.-Y. Lin, H.-J. Chen and R.-L. Sheu, Augmented lagrange primal-dual approach for generalized fractional programming problems, Journal of Industrial and Management Optimization, 4 (2013), 723-741. 

[16]

A. Nagih and G. Plateau, Problémes fractionnaires: Tour d'horizon sur les applications et méthodes de résolution, RAIRO Oper. Res., 33 (1999), 383-419. 

[17]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N. J., 1970.

[18]

A. Roubi, Method of centers for generalized fractional programming, Journal of Optimization Theory and Applications, 107 (2000), 123-143.  doi: 10.1023/A:1004660917684.

[19]

A. Roubi, Convergence of prox-regularization methods for generalized fractional programming, RAIRO Oper. Res., 36 (2002), 73-94.  doi: 10.1051/ro:2002006.

[20]

S. Schaible, Fractional Programming, in Handbook Global Optimization (eds. R. Horst and P. M. Pardalos), Kluwer, Dordrecht, (1995), 495{608

[21]

M. Sion, On General minimax theorems, Pacific Journal of Mathematics, 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171.

[22]

J. J. Strodiot, J. P. Crouzeix, J. A. Ferland and V. H. Nguyen, An inexact proximal point method for solving generalized fractional programs, Journal of Global Optimization, 42 (2008), 121-138.  doi: 10.1007/s10898-007-9270-x.

show all references

The authors would like to thank the referees for their valuable comments

References:
[1]

A. Addou and A. Roubi, Proximal-type methods with generalized bregman functions and applications to generalized fractional programming, Optimization, 59 (2010), 1085-1105.  doi: 10.1080/02331930903395857.

[2]

A. I. Barros, J. B. G. Frenk, S. Schaible and S. Zhang, A new algorithm for generalized fractional programs, Mathematical Programming, 72 (1996), 147-175.  doi: 10.1007/BF02592087.

[3]

A. I. Barros, J. B. G. Frenk, S. Schaible and S. Zhang, Using duality to solve generalized fractional programming problems, Journal of Global Optimization, 8 (1996), 139-170.  doi: 10.1007/BF00138690.

[4]

J. C. Bernard and J. A. Ferland, Convergence of interval-type algorithms for generalized fractional programming, Mathematical Programming, 43 (1989), 349-363.  doi: 10.1007/BF01582298.

[5]

J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming, SIAM Journal on Control and Optimization, 31 (1993), 1340-1359.  doi: 10.1137/0331063.

[6]

J. P. Crouzeix, J. A. Ferland and S. Schaible, Duality in generalized linear fractional programming, Mathematical Programming, 27 (1983), 342-354.  doi: 10.1007/BF02591908.

[7]

J. P. Crouzeix, J. A. Ferland and S. Schaible, An algorithm for generalized fractional programs, Journal of Optimization Theory and Applications, 47 (1985), 35-49.  doi: 10.1007/BF00941314.

[8]

J. P. Crouzeix, J. A. Ferland and S. Schaible, A note on an algorithm for generalized fractional programs, Journal of Optimization Theory and Applications, 50 (1986), 183-187.  doi: 10.1007/BF00938484.

[9]

J. P. Crouzeix and J. A. Ferland, Algorithms for generalized fractional programming, Mathematical Programming, 52 (1991), 191-207.  doi: 10.1007/BF01582887.

[10]

W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498.  doi: 10.1287/mnsc.13.7.492.

[11]

I. Ekeland and R. Temam, Analyse Convexe et Problémes Variationnels, Gauthier-Villars, Paris, Bruxelles, Montréal, 1976.

[12]

J. B. G. Frenk and S. Schaible, Fractional Programming, in ERIM Report Series, (Reference No. ERS-2004-074-LIS) (2004).

[13]

M. Gugat, Prox-regularization methods for generalized fractional programming, Journal of Optimization Theory and Applications, 99 (1998), 691-722.  doi: 10.1023/A:1021759318653.

[14]

O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization, 29 (1991), 403-419. 

[15]

J.-Y. Lin, H.-J. Chen and R.-L. Sheu, Augmented lagrange primal-dual approach for generalized fractional programming problems, Journal of Industrial and Management Optimization, 4 (2013), 723-741. 

[16]

A. Nagih and G. Plateau, Problémes fractionnaires: Tour d'horizon sur les applications et méthodes de résolution, RAIRO Oper. Res., 33 (1999), 383-419. 

[17]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N. J., 1970.

[18]

A. Roubi, Method of centers for generalized fractional programming, Journal of Optimization Theory and Applications, 107 (2000), 123-143.  doi: 10.1023/A:1004660917684.

[19]

A. Roubi, Convergence of prox-regularization methods for generalized fractional programming, RAIRO Oper. Res., 36 (2002), 73-94.  doi: 10.1051/ro:2002006.

[20]

S. Schaible, Fractional Programming, in Handbook Global Optimization (eds. R. Horst and P. M. Pardalos), Kluwer, Dordrecht, (1995), 495{608

[21]

M. Sion, On General minimax theorems, Pacific Journal of Mathematics, 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171.

[22]

J. J. Strodiot, J. P. Crouzeix, J. A. Ferland and V. H. Nguyen, An inexact proximal point method for solving generalized fractional programs, Journal of Global Optimization, 42 (2008), 121-138.  doi: 10.1007/s10898-007-9270-x.

Table 1.  The number of iterations and times with $n=5$, $m=5$, $p=5$
$\alpha$ Problems
1 2 3 4 5 6 7 8 9 10
10 It 62 143 81 4 27 2 9 763 86 114
T(s) 1.20 2.74 1.58 0.10 0.52 0.06 0.20 14.46 1.66 2.15
1 It 69 2 78 2 24 2 3 2 45 86
T(s) 1.25 0.05 1.47 0.05 0.45 0.06 0.07 0.05 0.85 1.58
10-1 It 69 2 78 2 24 2 2 2 31 80
T(s) 1.27 0.05 1.44 0.05 0.44 0.05 0.05 0.05 0.60 1.45
Alg[3] It 73 2 62 2 24 2 2 2 27 79
T(s) 1.18 0.04 1.09 0.04 0.39 0.05 0.04 0.04 0.46 1.27
Table Options

Download as excel

$\alpha$ Problems
1 2 3 4 5 6 7 8 9 10
10 It 62 143 81 4 27 2 9 763 86 114
T(s) 1.20 2.74 1.58 0.10 0.52 0.06 0.20 14.46 1.66 2.15
1 It 69 2 78 2 24 2 3 2 45 86
T(s) 1.25 0.05 1.47 0.05 0.45 0.06 0.07 0.05 0.85 1.58
10-1 It 69 2 78 2 24 2 2 2 31 80
T(s) 1.27 0.05 1.44 0.05 0.44 0.05 0.05 0.05 0.60 1.45
Alg[3] It 73 2 62 2 24 2 2 2 27 79
T(s) 1.18 0.04 1.09 0.04 0.39 0.05 0.04 0.04 0.46 1.27
Table 2.  The number of iterations and times with $n=10$, $m=10$, $p=5$
$\alpha$ Problems
1 2 3 4 5 6 7 8 9 10
10 It 328 165 62 50 21 91 6 13 31 225
T(s) 14.65 7.27 3.06 2.23 0.95 4.18 0.31 0.65 1.37 9.84
1 It 235 175 54 37 20 12 3 5 29 243
T(s) 10.38 7.73 2.47 1.70 0.87 0.57 0.22 0.24 1.26 10.59
10-1 It 232 234 53 37 19 5 3 5 29 244
T(s) 10.28 10.25 2.32 1.73 1.06 0.25 0.15 0.24 1.33 10.66
Alg[3] It 253 246 56 39 20 4 3 5 29 249
T(s) 4.70 4.48 1.01 0.68 0.35 0.09 0.06 0.11 0.51 4.50
Table Options

Download as excel

$\alpha$ Problems
1 2 3 4 5 6 7 8 9 10
10 It 328 165 62 50 21 91 6 13 31 225
T(s) 14.65 7.27 3.06 2.23 0.95 4.18 0.31 0.65 1.37 9.84
1 It 235 175 54 37 20 12 3 5 29 243
T(s) 10.38 7.73 2.47 1.70 0.87 0.57 0.22 0.24 1.26 10.59
10-1 It 232 234 53 37 19 5 3 5 29 244
T(s) 10.28 10.25 2.32 1.73 1.06 0.25 0.15 0.24 1.33 10.66
Alg[3] It 253 246 56 39 20 4 3 5 29 249
T(s) 4.70 4.48 1.01 0.68 0.35 0.09 0.06 0.11 0.51 4.50
Table 3.  The number of iterations and times with $n=20$, $m=10$, $p=5$
$\alpha$ Problems
1 2 3 4 5 6 7 8 9 10
10 It 87 26 260 24 151 70 50 11 113 62
T(s) 12.10 3.77 31.17 2.84 17.97 8.06 5.75 1.33 13.68 7.41
1 It 87 22 27 24 102 75 50 8 117 58
T(s) 11.38 2.67 3.40 2.78 12.00 8.83 5.87 0.95 14.14 6.77
10-1 It 87 22 8 24 102 75 50 8 122 58
T(s) 10.52 2.58 0.97 2.85 11.88 8.77 5.88 0.95 14.45 7.12
Alg [3] It 91 22 9 24 121 78 51 8 136 59
T(s) 2.01 0.47 0.23 0.54 2.41 1.68 1.05 0.19 3.09 1.24
Table Options

Download as excel

$\alpha$ Problems
1 2 3 4 5 6 7 8 9 10
10 It 87 26 260 24 151 70 50 11 113 62
T(s) 12.10 3.77 31.17 2.84 17.97 8.06 5.75 1.33 13.68 7.41
1 It 87 22 27 24 102 75 50 8 117 58
T(s) 11.38 2.67 3.40 2.78 12.00 8.83 5.87 0.95 14.14 6.77
10-1 It 87 22 8 24 102 75 50 8 122 58
T(s) 10.52 2.58 0.97 2.85 11.88 8.77 5.88 0.95 14.45 7.12
Alg [3] It 91 22 9 24 121 78 51 8 136 59
T(s) 2.01 0.47 0.23 0.54 2.41 1.68 1.05 0.19 3.09 1.24
Table 4.  The number of iterations and times with $n=50$, $m=10$, $p=5$
$\alpha$ Problems
1 2 3 4 5 6 7 8 9 10
10 It 35 50 35 26 78 19 39 91 42 109
T(s) 29.75 56.08 46.26 25.62 189.18 13.00 27.19 62.18 29.15 75.01
1 It 20 50 32 26 65 19 21 92 42 103
T(s) 19.07 44.95 29.71 77.36 155.83 12.93 14.44 62.76 29.08 70.70
10-1 It 20 50 32 26 64 19 21 92 42 103
T(s) 16.57 42.72 31.10 48.07 148.04 13.35 14.42 62.74 29.43 69.84
Alg [3] It 20 51 31 26 65 19 21 95 44 98
T(s) 0.76 2.94 1.18 2.25 2.04 0.59 0.61 3.18 1.41 3.37
Table Options

Download as excel

$\alpha$ Problems
1 2 3 4 5 6 7 8 9 10
10 It 35 50 35 26 78 19 39 91 42 109
T(s) 29.75 56.08 46.26 25.62 189.18 13.00 27.19 62.18 29.15 75.01
1 It 20 50 32 26 65 19 21 92 42 103
T(s) 19.07 44.95 29.71 77.36 155.83 12.93 14.44 62.76 29.08 70.70
10-1 It 20 50 32 26 64 19 21 92 42 103
T(s) 16.57 42.72 31.10 48.07 148.04 13.35 14.42 62.74 29.43 69.84
Alg [3] It 20 51 31 26 65 19 21 95 44 98
T(s) 0.76 2.94 1.18 2.25 2.04 0.59 0.61 3.18 1.41 3.37
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