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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

Hssaine Boualam and  Ahmed Roubi ,

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  October 2019 Early access  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.

Keywords: Minimax fractional programs, proximal point algorithm, bundle methods, quadratic programming.
Mathematics Subject Classification: Primary: 90C32, 90C25, 49K35; Secondary: 49M29, 49M37.
Citation: Hssaine Boualam, Ahmed Roubi. Dual algorithms based on the proximal bundle method for solving convex minimax fractional programs. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1897-1920. doi: 10.3934/jimo.2018128
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]

S. Addoune, M. El Haffari and A. Roubi, A proximal point algorithm for generalized fractional programs, Optimization, 66 (2017), 1495-1517.  doi: 10.1080/02331934.2017.1338698.

[3]

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.

[4]

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.

[5]

C. R. Bector, S. Chandra and M. K. Bector, Generalized fractional programming duality: A parametric approach, Journal of Optimization Theory and Applications, 60 (1989), 243-260.  doi: 10.1007/BF00940006.

[6]

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.

[7]

K. Boufi and A. Roubi, Prox-regularization of the dual method of centers for generalized fractional programs, To appear in Optimization Methods and Software. doi: 10.1080/10556788.2017.1392520.

[8]

K. Boufi and A. Roubi, Dual method of centers for solving generalized fractional programs, Journal of Global Optimization, 69 (2017), 387-426.  doi: 10.1007/s10898-017-0523-z.

[9]

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

[10]

R. Correa and C. Lemaréchal, Convergence of some algorithms for convex minimization, Mathematical Programming, 62 (1993), 261-275.  doi: 10.1007/BF01585170.

[11]

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

[12]

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.

[13]

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.

[14]

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.

[15]

M. El Haffari and A. Roubi, Convergence of a proximal algorithm for solving the dual of a generalized fractional program, RAIRO-Oper. Res., 51 (2017), 985-1004.  doi: 10.1051/ro/2017004.

[16]

M. El Haffari and A. Roubi, Prox-dual regularization algorithm for generalized fractional programs, Journal of Industrial and Management Optimization, 13 (2017), 1991-2013.  doi: 10.3934/jimo.2017028.

[17]

J. E. Falk, Maximization of signal-to-noise ratio in an optical filter, SIAM Appl. Math., 17 (1969), 582-592.  doi: 10.1137/0117055.

[18]

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

[19]

M. Fukushima, A descent algorithm for nonsmooth convex optimization, Mathematical Programming, 30 (1984), 163-175.  doi: 10.1007/BF02591883.

[20]

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

[21]

O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization, 29 (1991), 403-419.  doi: 10.1137/0329022.

[22]

J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms II, Springer-Verlag, 1993. doi: 10.1007/978-3-662-06409-2.

[23]

R. Jagannathan and S. Schaible, Duality in generalized fractional programming via Farkas lemma, Journal of Optimization Theory and Applications, 41 (1983), 417-424.  doi: 10.1007/BF00935361.

[24]

K. C. Kiwiel, An aggregate subgradient method for nonsmooth convex minimization, Mathematical Programming, 27 (1983), 320-341.  doi: 10.1007/BF02591907.

[25]

K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0074500.

[26]

K. C. Kiwiel, Proximity control in bundle methods for convex nondifferentiable minimization, Mathematical Programming, 46 (1990), 105-122.  doi: 10.1007/BF01585731.

[27]

C. Lemaréchal, Bundle methods in nonsmooth optimization, in Nonsmooth Optimization (eds. C. Lemaréchal and R. Mifflin), Pergamon Press, Oxford, 3 (1978), 79-102.

[28]

C. Lemaréchal, Constructing bundle methods for convex optimization, in Fermat Days 85: Mathematics for Optimization (eds. J. B. Hiriart-Urruty), North-Holland, Amsterdam, (1986), 201-240. doi: 10.1016/S0304-0208(08)72400-9.

[29]

M. Mäkelä, Survey of bundle methods for nonsmooth optimization, Optimization Methods and Software, 17 (2002), 1-29.  doi: 10.1080/10556780290027828.

[30]

B. Martinet, Régularisation d'inéquations variationnelles par approximation successives, Revue Française d'Informatique et Recherche Opérationnelle, 4 (1970), 154-158. 

[31]

R. Mifflin, An algorithm for constrained optimization with semismooth functions, Math. Oper. Res., 2 (1977), 191-207.  doi: 10.1287/moor.2.2.191.

[32]

R. Mifflin, A modification and extension of Lemaréchal's algorithm for nonsmooth minimization, Math. Programming Stud., 17 (1982), 77-90.  doi: 10.1007/BFb0120960.

[33]

J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299. 

[34]

A. Nagih and G. Plateau, Problèmes fractionnaires : tour d'horizon sur les applications et méthodes de résolution, RAIRO - Operations Research, 33 (1999), 383-419.  doi: 10.1051/ro:1999118.

[35]

B. T. Polyak, Introduction to Optimization, Translations Series in Mathematics and Engineering, Optimization Software, Inc. Publications Division, New York, 1987.

[36]

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

[37]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14 (1976), 877-898.  doi: 10.1137/0314056.

[38]

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

[39]

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

[40]

S. Schaible, Fractional programming, in Handbook Global Optimization (eds. R. Horst, PM Pardalos), Kluwer, Dordrecht, 2 (1995), 495-608.  doi: 10.1007/978-1-4615-2025-2.

[41]

H. Schramm and J. Zowe, A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results, SIAM Journal on Optimization, 2 (1992), 121-152.  doi: 10.1137/0802008.

[42]

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

[43]

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 a referee for his 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]

S. Addoune, M. El Haffari and A. Roubi, A proximal point algorithm for generalized fractional programs, Optimization, 66 (2017), 1495-1517.  doi: 10.1080/02331934.2017.1338698.

[3]

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.

[4]

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.

[5]

C. R. Bector, S. Chandra and M. K. Bector, Generalized fractional programming duality: A parametric approach, Journal of Optimization Theory and Applications, 60 (1989), 243-260.  doi: 10.1007/BF00940006.

[6]

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.

[7]

K. Boufi and A. Roubi, Prox-regularization of the dual method of centers for generalized fractional programs, To appear in Optimization Methods and Software. doi: 10.1080/10556788.2017.1392520.

[8]

K. Boufi and A. Roubi, Dual method of centers for solving generalized fractional programs, Journal of Global Optimization, 69 (2017), 387-426.  doi: 10.1007/s10898-017-0523-z.

[9]

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

[10]

R. Correa and C. Lemaréchal, Convergence of some algorithms for convex minimization, Mathematical Programming, 62 (1993), 261-275.  doi: 10.1007/BF01585170.

[11]

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

[12]

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.

[13]

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.

[14]

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.

[15]

M. El Haffari and A. Roubi, Convergence of a proximal algorithm for solving the dual of a generalized fractional program, RAIRO-Oper. Res., 51 (2017), 985-1004.  doi: 10.1051/ro/2017004.

[16]

M. El Haffari and A. Roubi, Prox-dual regularization algorithm for generalized fractional programs, Journal of Industrial and Management Optimization, 13 (2017), 1991-2013.  doi: 10.3934/jimo.2017028.

[17]

J. E. Falk, Maximization of signal-to-noise ratio in an optical filter, SIAM Appl. Math., 17 (1969), 582-592.  doi: 10.1137/0117055.

[18]

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

[19]

M. Fukushima, A descent algorithm for nonsmooth convex optimization, Mathematical Programming, 30 (1984), 163-175.  doi: 10.1007/BF02591883.

[20]

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

[21]

O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization, 29 (1991), 403-419.  doi: 10.1137/0329022.

[22]

J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms II, Springer-Verlag, 1993. doi: 10.1007/978-3-662-06409-2.

[23]

R. Jagannathan and S. Schaible, Duality in generalized fractional programming via Farkas lemma, Journal of Optimization Theory and Applications, 41 (1983), 417-424.  doi: 10.1007/BF00935361.

[24]

K. C. Kiwiel, An aggregate subgradient method for nonsmooth convex minimization, Mathematical Programming, 27 (1983), 320-341.  doi: 10.1007/BF02591907.

[25]

K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0074500.

[26]

K. C. Kiwiel, Proximity control in bundle methods for convex nondifferentiable minimization, Mathematical Programming, 46 (1990), 105-122.  doi: 10.1007/BF01585731.

[27]

C. Lemaréchal, Bundle methods in nonsmooth optimization, in Nonsmooth Optimization (eds. C. Lemaréchal and R. Mifflin), Pergamon Press, Oxford, 3 (1978), 79-102.

[28]

C. Lemaréchal, Constructing bundle methods for convex optimization, in Fermat Days 85: Mathematics for Optimization (eds. J. B. Hiriart-Urruty), North-Holland, Amsterdam, (1986), 201-240. doi: 10.1016/S0304-0208(08)72400-9.

[29]

M. Mäkelä, Survey of bundle methods for nonsmooth optimization, Optimization Methods and Software, 17 (2002), 1-29.  doi: 10.1080/10556780290027828.

[30]

B. Martinet, Régularisation d'inéquations variationnelles par approximation successives, Revue Française d'Informatique et Recherche Opérationnelle, 4 (1970), 154-158. 

[31]

R. Mifflin, An algorithm for constrained optimization with semismooth functions, Math. Oper. Res., 2 (1977), 191-207.  doi: 10.1287/moor.2.2.191.

[32]

R. Mifflin, A modification and extension of Lemaréchal's algorithm for nonsmooth minimization, Math. Programming Stud., 17 (1982), 77-90.  doi: 10.1007/BFb0120960.

[33]

J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299. 

[34]

A. Nagih and G. Plateau, Problèmes fractionnaires : tour d'horizon sur les applications et méthodes de résolution, RAIRO - Operations Research, 33 (1999), 383-419.  doi: 10.1051/ro:1999118.

[35]

B. T. Polyak, Introduction to Optimization, Translations Series in Mathematics and Engineering, Optimization Software, Inc. Publications Division, New York, 1987.

[36]

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

[37]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14 (1976), 877-898.  doi: 10.1137/0314056.

[38]

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

[39]

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

[40]

S. Schaible, Fractional programming, in Handbook Global Optimization (eds. R. Horst, PM Pardalos), Kluwer, Dordrecht, 2 (1995), 495-608.  doi: 10.1007/978-1-4615-2025-2.

[41]

H. Schramm and J. Zowe, A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results, SIAM Journal on Optimization, 2 (1992), 121-152.  doi: 10.1137/0802008.

[42]

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

[43]

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.  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
Table Options

Download as excel

$\bf{n=10}$ $\bf{m=10}$ $\bf{p=5}$
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
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
Table Options

Download as excel

$\bf{n=20}$ $\bf{m=10}$ $\bf{p=10}$
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
[1]

Hssaine Boualam, Ahmed Roubi. Augmented Lagrangian dual for nonconvex minimax fractional programs and proximal bundle algorithms for its resolution. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022100
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