-
Previous Article
A mean-reverting currency model with floating interest rates in uncertain environment
- JIMO Home
- This Issue
-
Next Article
Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method
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 |
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.
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. Google Scholar |
[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
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. Google Scholar |
[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. |
Algo 5 | Algo [43] | ||||||||||||
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 |
Algo 5 | Algo [43] | ||||||||||||
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 |
Algo 5 | Algo [43] | ||||||||||||
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 |
Algo 5 | Algo [43] | ||||||||||||
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] |
Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565 |
[2] |
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 |
[3] |
María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 |
[4] |
Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141 |
[5] |
Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881 |
[6] |
Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 |
[7] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[8] |
Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781 |
[9] |
Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024 |
[10] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[11] |
Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]