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
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. Google Scholar

[2]

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

[3]

A. I. BarrosJ. B. G. FrenkS. Schaible and S. Zhang, A new algorithm for generalized fractional programs, Mathematical Programming, 72 (1996), 147-175. doi: 10.1007/BF02592087. Google Scholar

[4]

A. I. BarrosJ. B. G. FrenkS. Schaible and S. Zhang, Using duality to solve generalized fractional programming problems, Journal of Global Optimization, 8 (1996), 139-170. doi: 10.1007/BF00138690. Google Scholar

[5]

C. R. BectorS. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[11]

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

[12]

J. P. CrouzeixJ. A. Ferland and S. Schaible, Duality in generalized linear fractional programming, Mathematical Programming, 27 (1983), 342-354. doi: 10.1007/BF02591908. Google Scholar

[13]

J. P. CrouzeixJ. 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. Google Scholar

[14]

J. P. CrouzeixJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[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. Google Scholar

[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. Google Scholar

[33]

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

[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. Google Scholar

[35]

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

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[42]

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

[43]

J. J. StrodiotJ. P. CrouzeixJ. 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. Google Scholar

show all references

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. Google Scholar

[2]

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

[3]

A. I. BarrosJ. B. G. FrenkS. Schaible and S. Zhang, A new algorithm for generalized fractional programs, Mathematical Programming, 72 (1996), 147-175. doi: 10.1007/BF02592087. Google Scholar

[4]

A. I. BarrosJ. B. G. FrenkS. Schaible and S. Zhang, Using duality to solve generalized fractional programming problems, Journal of Global Optimization, 8 (1996), 139-170. doi: 10.1007/BF00138690. Google Scholar

[5]

C. R. BectorS. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[11]

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

[12]

J. P. CrouzeixJ. A. Ferland and S. Schaible, Duality in generalized linear fractional programming, Mathematical Programming, 27 (1983), 342-354. doi: 10.1007/BF02591908. Google Scholar

[13]

J. P. CrouzeixJ. 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. Google Scholar

[14]

J. P. CrouzeixJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[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. Google Scholar

[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. Google Scholar

[33]

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

[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. Google Scholar

[35]

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

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[42]

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

[43]

J. J. StrodiotJ. P. CrouzeixJ. 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. Google Scholar

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
$\bf{n=10}$ $\bf{m=10}$ $\bf{p=5}$
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
$\bf{n=20}$ $\bf{m=10}$ $\bf{p=10}$
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]

Chunming Tang, Jinbao Jian, Guoyin Li. A proximal-projection partial bundle method for convex constrained minimax problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 757-774. doi: 10.3934/jimo.2018069

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