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Augmented Lagrange primal-dual approach for generalized fractional programming problems
1. | Department of Applied Mathematics, No.300 Syuefu Rd., Chiayi City 60004, Taiwan |
2. | Department of Mathematics, No.1, University Road, Tainan City 701, Taiwan, Taiwan |
References:
[1] |
M. Avriel, E. Diewert, S. Schaible and I. Zang, "Generalized Concavity,'' Society for Industrial and Applied Mathematics, 2010. |
[2] |
D. P. Bertsekas, "Constrained Optimization and Lagrange Multiplier Methods,'' Computer Science and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982. |
[3] |
D. P. Bertsekas, "Convex Analysis and Optimization,'' With Angelia Nedié and Asuman E. Ozdaglar, Athena Scientific, Belmont, MA, 2003. |
[4] |
J. C. Bernard and J. A. Ferland, Convergence of interval-type algorithms for generalized fractional programming, Math. Programming, 43 (1989), 349-363.
doi: 10.1007/BF01582298. |
[5] |
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. |
[6] |
I. Barrodale, M. J. D. Powell and F. D. K. Roberts, The differential correction algorithm for rational $l_{\infty}$-approximation, SIAM J. Numer. Anal., 9 (1972), 493-504.
doi: 10.1137/0709044. |
[7] |
A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Naval Research Logistics Quarterly, 9 (1962), 181-186.
doi: 10.1002/nav.3800090303. |
[8] |
J. P. Crouzeix and J. A. Ferland, Algorithms for generalized fractional programming, Mathematical Programming, 52 (1991), 191-207.
doi: 10.1007/BF01582887. |
[9] |
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. |
[10] |
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. |
[11] |
S.-H. Chu, "Optimal Resources Allocation for a Cognitive Network," Master's thesis, National Cheng Kung University, Taiwan, ROC, 2009.
doi: 10.1007/s11277-012-0657-8. |
[12] |
B. D. Craven, "Fractional Programming,'' Sigma Series in Applied Mathematics, 4, Heldermann Verlag, Berlin, 1988. |
[13] |
H. J. Chen, S. Schaible and R. L. Sheu, Generic algorithm for generalized fractional programming, J. Optim. Theory Appl., 141 (2009), 93-105.
doi: 10.1007/s10957-008-9499-7. |
[14] |
W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498.
doi: 10.1287/mnsc.13.7.492. |
[15] |
C. A. Floudas and P. M. Pardalos, eds., "Encyclopedia of Optimization,'' Second edition, Springer, 2009. |
[16] |
N. Hadjisavvas, S. Komlósi and S. Schaible, eds., "Handbook of generalized convexity and generalized monotonicity,'' Nonconvex Optimization and its Applications, 76, Springer-Verlag, New York, 2005.
doi: 10.1007/b101428. |
[17] |
J. B. Hiriart-Urruty and C. Lemarechal, "Convex Analysis and Minimization Algorithm,'' Springer-Verlag, Berlin, 1994. |
[18] |
R. Jagannathan, On projective representations of finite abelian groups, in "Number Theory" (Ootacamund, 1984), Lecture Notes in Math., 1122, Springer, Berlin, (1985), 130-139.
doi: 10.1007/BFb0075756. |
[19] |
J. von Neumann, A model of general economic equilibrium, The Review of Economic Studies, 13 (1945), 1-9.
doi: 10.2307/2296111. |
[20] |
E. Polak, J. E. Higgins and D. Q. Mayne, A barrier function method for minimax problems, Math. Program., 54 (1992), 155-176.
doi: 10.1007/BF01586049. |
[21] |
R. T. Rockafellar, "Convex Analysis,'' Reprint of the 1970 original, Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 1997. |
[22] |
S. Schaible, Fractional programming. I. Duality, Management Science, 22 (1976), 858-867. |
[23] |
S. Schaible, Multi-ratio fractional programming-a survey, in "Optimization, Parallel Processing and Applications'' (Oberwolfach, 1987 and Karlsruhe, 1987), Lecture Notes in Econom. and Math. Systems, 304, Springer, Berlin, (1988), 57-66.
doi: 10.1007/978-3-642-46631-1_7. |
[24] |
S. Schaible and J. Shi, Recent developments in fractional programming: Single-ratio and max-min case, in "Nonlinear Analysis and Convex Analysis," Yokohama Publ., Yokohama, (2004), 493-506. |
[25] |
R.-L. Sheu and J.-Y. Lin, Solving continuous min-max problems by an iterative entropic regularization method, J. Optim. Theory Appl., 121 (2004), 597-612.
doi: 10.1023/B:JOTA.0000037605.19435.63. |
[26] |
M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.
doi: 10.2140/pjm.1958.8.171. |
[27] |
I. M. Stancu-Minasian, Bibliography of fractional programming, 1960-1976, Pure Appl. Math. Sci., 13 (1981), 35-69. |
[28] |
I. M. Stancu-Minasian, A second bibliography of fractional programming: 1977-1981, Pure Appl. Math. Sci., 17 (1983), 87-102. |
[29] |
I. M. Stancu-Minasian, A third bibliography of fractional programming, Pure Appl. Math. Sci., 22 (1985), 109-122. |
[30] |
I. M. Stancu-Minasian, A fourth bibliography of fractional programming, Optimization, 23 (1992), 53-71.
doi: 10.1080/02331939208843744. |
[31] |
I. M. Stancu-Minasian, A fifth bibliography of fractional programming, Dedicated to the memory of Professor Karl-Heinz Elster, Optimization, 45 (1999), 343-367.
doi: 10.1080/02331939908844438. |
[32] |
I. M. Stancu-Minasian, A sixth bibliography of fractional programming, Optimization, 55 (2006), 405-428.
doi: 10.1080/02331930600819613. |
show all references
References:
[1] |
M. Avriel, E. Diewert, S. Schaible and I. Zang, "Generalized Concavity,'' Society for Industrial and Applied Mathematics, 2010. |
[2] |
D. P. Bertsekas, "Constrained Optimization and Lagrange Multiplier Methods,'' Computer Science and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982. |
[3] |
D. P. Bertsekas, "Convex Analysis and Optimization,'' With Angelia Nedié and Asuman E. Ozdaglar, Athena Scientific, Belmont, MA, 2003. |
[4] |
J. C. Bernard and J. A. Ferland, Convergence of interval-type algorithms for generalized fractional programming, Math. Programming, 43 (1989), 349-363.
doi: 10.1007/BF01582298. |
[5] |
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. |
[6] |
I. Barrodale, M. J. D. Powell and F. D. K. Roberts, The differential correction algorithm for rational $l_{\infty}$-approximation, SIAM J. Numer. Anal., 9 (1972), 493-504.
doi: 10.1137/0709044. |
[7] |
A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Naval Research Logistics Quarterly, 9 (1962), 181-186.
doi: 10.1002/nav.3800090303. |
[8] |
J. P. Crouzeix and J. A. Ferland, Algorithms for generalized fractional programming, Mathematical Programming, 52 (1991), 191-207.
doi: 10.1007/BF01582887. |
[9] |
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. |
[10] |
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. |
[11] |
S.-H. Chu, "Optimal Resources Allocation for a Cognitive Network," Master's thesis, National Cheng Kung University, Taiwan, ROC, 2009.
doi: 10.1007/s11277-012-0657-8. |
[12] |
B. D. Craven, "Fractional Programming,'' Sigma Series in Applied Mathematics, 4, Heldermann Verlag, Berlin, 1988. |
[13] |
H. J. Chen, S. Schaible and R. L. Sheu, Generic algorithm for generalized fractional programming, J. Optim. Theory Appl., 141 (2009), 93-105.
doi: 10.1007/s10957-008-9499-7. |
[14] |
W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498.
doi: 10.1287/mnsc.13.7.492. |
[15] |
C. A. Floudas and P. M. Pardalos, eds., "Encyclopedia of Optimization,'' Second edition, Springer, 2009. |
[16] |
N. Hadjisavvas, S. Komlósi and S. Schaible, eds., "Handbook of generalized convexity and generalized monotonicity,'' Nonconvex Optimization and its Applications, 76, Springer-Verlag, New York, 2005.
doi: 10.1007/b101428. |
[17] |
J. B. Hiriart-Urruty and C. Lemarechal, "Convex Analysis and Minimization Algorithm,'' Springer-Verlag, Berlin, 1994. |
[18] |
R. Jagannathan, On projective representations of finite abelian groups, in "Number Theory" (Ootacamund, 1984), Lecture Notes in Math., 1122, Springer, Berlin, (1985), 130-139.
doi: 10.1007/BFb0075756. |
[19] |
J. von Neumann, A model of general economic equilibrium, The Review of Economic Studies, 13 (1945), 1-9.
doi: 10.2307/2296111. |
[20] |
E. Polak, J. E. Higgins and D. Q. Mayne, A barrier function method for minimax problems, Math. Program., 54 (1992), 155-176.
doi: 10.1007/BF01586049. |
[21] |
R. T. Rockafellar, "Convex Analysis,'' Reprint of the 1970 original, Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 1997. |
[22] |
S. Schaible, Fractional programming. I. Duality, Management Science, 22 (1976), 858-867. |
[23] |
S. Schaible, Multi-ratio fractional programming-a survey, in "Optimization, Parallel Processing and Applications'' (Oberwolfach, 1987 and Karlsruhe, 1987), Lecture Notes in Econom. and Math. Systems, 304, Springer, Berlin, (1988), 57-66.
doi: 10.1007/978-3-642-46631-1_7. |
[24] |
S. Schaible and J. Shi, Recent developments in fractional programming: Single-ratio and max-min case, in "Nonlinear Analysis and Convex Analysis," Yokohama Publ., Yokohama, (2004), 493-506. |
[25] |
R.-L. Sheu and J.-Y. Lin, Solving continuous min-max problems by an iterative entropic regularization method, J. Optim. Theory Appl., 121 (2004), 597-612.
doi: 10.1023/B:JOTA.0000037605.19435.63. |
[26] |
M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.
doi: 10.2140/pjm.1958.8.171. |
[27] |
I. M. Stancu-Minasian, Bibliography of fractional programming, 1960-1976, Pure Appl. Math. Sci., 13 (1981), 35-69. |
[28] |
I. M. Stancu-Minasian, A second bibliography of fractional programming: 1977-1981, Pure Appl. Math. Sci., 17 (1983), 87-102. |
[29] |
I. M. Stancu-Minasian, A third bibliography of fractional programming, Pure Appl. Math. Sci., 22 (1985), 109-122. |
[30] |
I. M. Stancu-Minasian, A fourth bibliography of fractional programming, Optimization, 23 (1992), 53-71.
doi: 10.1080/02331939208843744. |
[31] |
I. M. Stancu-Minasian, A fifth bibliography of fractional programming, Dedicated to the memory of Professor Karl-Heinz Elster, Optimization, 45 (1999), 343-367.
doi: 10.1080/02331939908844438. |
[32] |
I. M. Stancu-Minasian, A sixth bibliography of fractional programming, Optimization, 55 (2006), 405-428.
doi: 10.1080/02331930600819613. |
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