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October  2013, 9(4): 723-741. doi: 10.3934/jimo.2013.9.723

Augmented Lagrange primal-dual approach for generalized fractional programming problems

Jen-Yen Lin 1, , Hui-Ju Chen 2, and  Ruey-Lin Sheu 2,

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

Received  August 2011 Revised  March 2013 Published  August 2013

In this paper, we propose a primal-dual approach for solving the generalized fractional programming problem. The outer iteration of the algorithm is a variant of interval-type Dinkelbach algorithm, while the augmented Lagrange method is adopted for solving the inner min-max subproblems. This is indeed a very unique feature of the paper because almost all Dinkelbach-type algorithms in the literature addressed only the outer iteration, while leaving the issue of how to practically solve a sequence of min-max subproblems untouched. The augmented Lagrange method attaches a set of artificial variables as well as their corresponding Lagrange multipliers to the min-max subproblem. As a result, both the primal and the dual information is available for updating the iterate points and the min-max subproblem is then reduced to a sequence of minimization problems. Numerical experiments show that the primal-dual approach can achieve a better precision in fewer iterations.
Keywords: exponential penalty method., Generalized fractional program, augmented Lagrange method, Dinkelbach-type algorithm.
Mathematics Subject Classification: Primary: 49J35, 49N15, 49M37; Secondary: 49K3.
Citation: Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial & Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723
References:
[1]

M. Avriel, E. Diewert, S. Schaible and I. Zang, "Generalized Concavity,'' Society for Industrial and Applied Mathematics, 2010. Google Scholar

[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.  Google Scholar

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D. P. Bertsekas, "Convex Analysis and Optimization,'' With Angelia Nedié and Asuman E. Ozdaglar, Athena Scientific, Belmont, MA, 2003.  Google Scholar

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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.  Google Scholar

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A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Naval Research Logistics Quarterly, 9 (1962), 181-186. doi: 10.1002/nav.3800090303.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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B. D. Craven, "Fractional Programming,'' Sigma Series in Applied Mathematics, 4, Heldermann Verlag, Berlin, 1988.  Google Scholar

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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.  Google Scholar

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W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498. doi: 10.1287/mnsc.13.7.492.  Google Scholar

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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.  Google Scholar

[19]

J. von Neumann, A model of general economic equilibrium, The Review of Economic Studies, 13 (1945), 1-9. doi: 10.2307/2296111.  Google Scholar

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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.  Google Scholar

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R. T. Rockafellar, "Convex Analysis,'' Reprint of the 1970 original, Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 1997.  Google Scholar

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S. Schaible, Fractional programming. I. Duality, Management Science, 22 (1976), 858-867.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

I. M. Stancu-Minasian, Bibliography of fractional programming, 1960-1976, Pure Appl. Math. Sci., 13 (1981), 35-69.  Google Scholar

[28]

I. M. Stancu-Minasian, A second bibliography of fractional programming: 1977-1981, Pure Appl. Math. Sci., 17 (1983), 87-102.  Google Scholar

[29]

I. M. Stancu-Minasian, A third bibliography of fractional programming, Pure Appl. Math. Sci., 22 (1985), 109-122.  Google Scholar

[30]

I. M. Stancu-Minasian, A fourth bibliography of fractional programming, Optimization, 23 (1992), 53-71. doi: 10.1080/02331939208843744.  Google Scholar

[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.  Google Scholar

[32]

I. M. Stancu-Minasian, A sixth bibliography of fractional programming, Optimization, 55 (2006), 405-428. doi: 10.1080/02331930600819613.  Google Scholar

show all references

References:
[1]

M. Avriel, E. Diewert, S. Schaible and I. Zang, "Generalized Concavity,'' Society for Industrial and Applied Mathematics, 2010. Google Scholar

[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.  Google Scholar

[3]

D. P. Bertsekas, "Convex Analysis and Optimization,'' With Angelia Nedié and Asuman E. Ozdaglar, Athena Scientific, Belmont, MA, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

B. D. Craven, "Fractional Programming,'' Sigma Series in Applied Mathematics, 4, Heldermann Verlag, Berlin, 1988.  Google Scholar

[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.  Google Scholar

[14]

W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498. doi: 10.1287/mnsc.13.7.492.  Google Scholar

[15]

C. A. Floudas and P. M. Pardalos, eds., "Encyclopedia of Optimization,'' Second edition, Springer, 2009. Google Scholar

[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.  Google Scholar

[17]

J. B. Hiriart-Urruty and C. Lemarechal, "Convex Analysis and Minimization Algorithm,'' Springer-Verlag, Berlin, 1994. Google Scholar

[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.  Google Scholar

[19]

J. von Neumann, A model of general economic equilibrium, The Review of Economic Studies, 13 (1945), 1-9. doi: 10.2307/2296111.  Google Scholar

[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.  Google Scholar

[21]

R. T. Rockafellar, "Convex Analysis,'' Reprint of the 1970 original, Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[22]

S. Schaible, Fractional programming. I. Duality, Management Science, 22 (1976), 858-867.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

I. M. Stancu-Minasian, Bibliography of fractional programming, 1960-1976, Pure Appl. Math. Sci., 13 (1981), 35-69.  Google Scholar

[28]

I. M. Stancu-Minasian, A second bibliography of fractional programming: 1977-1981, Pure Appl. Math. Sci., 17 (1983), 87-102.  Google Scholar

[29]

I. M. Stancu-Minasian, A third bibliography of fractional programming, Pure Appl. Math. Sci., 22 (1985), 109-122.  Google Scholar

[30]

I. M. Stancu-Minasian, A fourth bibliography of fractional programming, Optimization, 23 (1992), 53-71. doi: 10.1080/02331939208843744.  Google Scholar

[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.  Google Scholar

[32]

I. M. Stancu-Minasian, A sixth bibliography of fractional programming, Optimization, 55 (2006), 405-428. doi: 10.1080/02331930600819613.  Google Scholar

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