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

## Prox-dual regularization algorithm for generalized fractional programs

 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.

Citation: Mostafa El Haffari, Ahmed Roubi. Prox-dual regularization algorithm for generalized fractional programs. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1991-2013. doi: 10.3934/jimo.2017028
##### References:

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##### References:
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
 $\alpha$ 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
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
 $\alpha$ 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
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
 $\alpha$ 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
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
 $\alpha$ 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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