# American Institute of Mathematical Sciences

October  2019, 15(4): 1897-1920. doi: 10.3934/jimo.2018128

## Dual algorithms based on the proximal bundle method for solving convex minimax fractional programs

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

* Corresponding author: Ahmed Roubi

The authors would like to thank a referee for his valuable comments

Received  January 2018 Revised  March 2018 Published  August 2018

In this work, we propose an approximating scheme based on the proximal point algorithm, for solving generalized fractional programs (GFP) by their continuous reformulation, also known to as partial dual counterparts of GFP. Bundle dual algorithms are then derived from this scheme. We prove the convergence and the rate of convergence of these algorithms. As for dual algorithms, the proposed methods generate a sequence of values that converges from below to the minimal value of $(P)$, and a sequence of approximate solutions that converges to a solution of the dual problem. For certain classes of problems, the convergence is at least linear.

Citation: Hssaine Boualam, Ahmed Roubi. Dual algorithms based on the proximal bundle method for solving convex minimax fractional programs. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1897-1920. doi: 10.3934/jimo.2018128
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##### References:
Results for Algorithm 5 and Algorithm [43] with $n = 10$, $m = 10$ and $p = 5$.
 $\bf{n=10}$ $\bf{m=10}$ $\bf{p=5}$ Algo 5 Algo [43] $\bf{ \pmb{\mathsf{ β}}_k}$ $c$ 0.01 0.05 0.1 0.5 0.9 0.01 0.05 0.1 0.5 0.9 0.01 Av. IT 13.1 11.2 10.6 9.7 6.8 8.1 8.8 7.8 5.8 4.6 Av. QP 137.7 102 95.8 89.8 73.6 81.3 88.3 78.7 62.6 48.8 Av. T(s) 21.6 14.6 13.7 13.5 11.7 12 12.7 11.6 10.4 8.2 0.5 Av. IT 13.1 12.9 12.2 9.9 7.2 8.7 8.8 7.4 6 4.5 Av. QP 81.6 84.5 75.7 66.2 56.4 67.4 69.6 55 47.9 40 Av. T(s) 8.8 9.6 8.8 8.4 7.3 8.8 9.1 7.7 7.3 6.3 1 Av. IT 13.8 13 10.7 10.3 7.1 8.6 7.6 7.4 5.3 4.7 Av. QP 76.2 66.5 53.1 60.6 47.1 53.9 50.1 51.4 38.4 37.1 Av. T(s) 8.3 7.1 5.5 7.7 5.7 6.8 6.6 6.8 5.5 5.6 5 Av. IT 12.8 11.3 10.8 12.1 10.5 9.4 8.3 8.4 9 8.3 Av. QP 43.6 37.6 34.8 45.5 48.1 39.7 35.7 36.2 46 47.7 Av. T(s) 4.5 3.9 3.6 4.6 4.9 4.5 4.2 4.3 5.4 5.7 10 Av. IT 12.5 11.8 11.5 12 13.3 12.8 12.3 11.9 13.4 13.4 Av. QP 34.7 30 27.7 34.1 49.8 45.1 42.7 41.7 57.2 61.3 Av. T(s) 3.9 3.3 3.1 3.7 5 4.4 4.2 4.2 5.5 6.2
 $\bf{n=10}$ $\bf{m=10}$ $\bf{p=5}$ Algo 5 Algo [43] $\bf{ \pmb{\mathsf{ β}}_k}$ $c$ 0.01 0.05 0.1 0.5 0.9 0.01 0.05 0.1 0.5 0.9 0.01 Av. IT 13.1 11.2 10.6 9.7 6.8 8.1 8.8 7.8 5.8 4.6 Av. QP 137.7 102 95.8 89.8 73.6 81.3 88.3 78.7 62.6 48.8 Av. T(s) 21.6 14.6 13.7 13.5 11.7 12 12.7 11.6 10.4 8.2 0.5 Av. IT 13.1 12.9 12.2 9.9 7.2 8.7 8.8 7.4 6 4.5 Av. QP 81.6 84.5 75.7 66.2 56.4 67.4 69.6 55 47.9 40 Av. T(s) 8.8 9.6 8.8 8.4 7.3 8.8 9.1 7.7 7.3 6.3 1 Av. IT 13.8 13 10.7 10.3 7.1 8.6 7.6 7.4 5.3 4.7 Av. QP 76.2 66.5 53.1 60.6 47.1 53.9 50.1 51.4 38.4 37.1 Av. T(s) 8.3 7.1 5.5 7.7 5.7 6.8 6.6 6.8 5.5 5.6 5 Av. IT 12.8 11.3 10.8 12.1 10.5 9.4 8.3 8.4 9 8.3 Av. QP 43.6 37.6 34.8 45.5 48.1 39.7 35.7 36.2 46 47.7 Av. T(s) 4.5 3.9 3.6 4.6 4.9 4.5 4.2 4.3 5.4 5.7 10 Av. IT 12.5 11.8 11.5 12 13.3 12.8 12.3 11.9 13.4 13.4 Av. QP 34.7 30 27.7 34.1 49.8 45.1 42.7 41.7 57.2 61.3 Av. T(s) 3.9 3.3 3.1 3.7 5 4.4 4.2 4.2 5.5 6.2
Results for Algorithm 5 and Algorithm [43] with $n = 20$, $m = 10$ and $p = 10$.
 $\bf{n=20}$ $\bf{m=10}$ $\bf{p=10}$ Algo 5 Algo [43] $\bf{ \pmb{\mathsf{ β}}_k}$ $c$ 0.01 0.05 0.1 0.5 0.9 0.01 0.05 0.1 0.5 0.9 0.01 Av. IT 13.3 12.9 11.5 10.8 8.8 7.9 8.4 7.5 5.2 4.4 Av. QP 119.7 117.4 90.9 95.3 87.9 97.8 104.2 93.9 64.6 59.4 Av. T(s) 34.2 33.9 25.7 27.5 25.2 16.5 17.7 16.3 12 11.4 0.5 Av. IT 12.6 12.4 11.2 12.3 9.5 8 8.8 8.1 5.3 4.5 Av. QP 71.3 65.5 59.6 71.8 64.1 78.1 93.5 81.6 56.4 47.2 Av. T(s) 17.2 16.7 15 19.5 17.5 12.4 14.6 13.1 10 8.9 1 Av. IT 13.4 13.1 13.1 12.3 10.4 9.6 8.8 8.2 5.2 4.9 Av. QP 64.2 63.6 59.3 59.4 58.5 83.2 75 72.7 50.9 43 Av. T(s) 16.6 16.5 15.5 16.6 16 13.1 11.8 11.5 9 7.8 5 Av. IT 19.5 17.3 16.7 17.2 20.3 9.5 9.3 9.5 7.7 8.1 Av. QP 57.1 45.1 41 47.2 64 45.3 45.7 49.8 42.1 54.3 Av. T(s) 14.1 11.1 10.6 11.9 17.2 6.8 6.9 7.3 6.7 8.6 10 Av. IT 30.1 30 30.2 30.7 33.4 12.7 12.7 11.7 13.5 12.2 Av. QP 59.7 57.7 56.7 59.9 73.1 52.2 52.7 48.1 64.8 71.4 Av. T(s) 15.4 15.2 14.8 15.7 20.5 6.9 6.9 6.4 8.2 9.4
 $\bf{n=20}$ $\bf{m=10}$ $\bf{p=10}$ Algo 5 Algo [43] $\bf{ \pmb{\mathsf{ β}}_k}$ $c$ 0.01 0.05 0.1 0.5 0.9 0.01 0.05 0.1 0.5 0.9 0.01 Av. IT 13.3 12.9 11.5 10.8 8.8 7.9 8.4 7.5 5.2 4.4 Av. QP 119.7 117.4 90.9 95.3 87.9 97.8 104.2 93.9 64.6 59.4 Av. T(s) 34.2 33.9 25.7 27.5 25.2 16.5 17.7 16.3 12 11.4 0.5 Av. IT 12.6 12.4 11.2 12.3 9.5 8 8.8 8.1 5.3 4.5 Av. QP 71.3 65.5 59.6 71.8 64.1 78.1 93.5 81.6 56.4 47.2 Av. T(s) 17.2 16.7 15 19.5 17.5 12.4 14.6 13.1 10 8.9 1 Av. IT 13.4 13.1 13.1 12.3 10.4 9.6 8.8 8.2 5.2 4.9 Av. QP 64.2 63.6 59.3 59.4 58.5 83.2 75 72.7 50.9 43 Av. T(s) 16.6 16.5 15.5 16.6 16 13.1 11.8 11.5 9 7.8 5 Av. IT 19.5 17.3 16.7 17.2 20.3 9.5 9.3 9.5 7.7 8.1 Av. QP 57.1 45.1 41 47.2 64 45.3 45.7 49.8 42.1 54.3 Av. T(s) 14.1 11.1 10.6 11.9 17.2 6.8 6.9 7.3 6.7 8.6 10 Av. IT 30.1 30 30.2 30.7 33.4 12.7 12.7 11.7 13.5 12.2 Av. QP 59.7 57.7 56.7 59.9 73.1 52.2 52.7 48.1 64.8 71.4 Av. T(s) 15.4 15.2 14.8 15.7 20.5 6.9 6.9 6.4 8.2 9.4
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