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Prox-dual regularization algorithm for generalized fractional programs
1. | Laboratoire MISI, Faculté des Sciences et Techniques, Univ. Hassan 1,26000 Settat, Morocco |
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.
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. |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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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