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

Hssaine Boualam; Ahmed Roubi

doi:10.3934/jimo.2018128

Article Contents

2019, Volume 15, Issue 4: 1897-1920. Doi: 10.3934/jimo.2018128

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

Hssaine Boualam and
Ahmed Roubi^,

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

* Corresponding author: Ahmed Roubi
* Corresponding author: Ahmed Roubi

Received: January 2018

Revised: March 2018

Early access: August 2018

Published: July 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 90C32, 90C25, 49K35; Secondary: 49M29, 49M37.

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Table 1. 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

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Table 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

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