Efficient algorithms and implementations for optimizing the sum of linear fractional functions, with applications

Journal of Combinatorial Optimization

Volumn 9, Issue 1, 2005, Pages 69-90

  Chen, Danny Z ^a   Daescu, Ovidiu ^b   Dai, Yang ^c   Katoh, Naoki ^d   Wu, Xiaodong ^e   Xu, Jinhui ^f  

^a University of Notre Dame   (United States)

^b The University of Texas at Dallas   (United States)

^c University of Illinois at Chicago   (United States)

^d KYOTO UNIVERSITY   (Japan)

^e UNIVERSITY OF TEXAS PAN AMERICAN   (United States)

^f State University of New York   (United States)

Author keywords

Combinatorial optimization; Computational geometry; Parametric linear programming; Robustness; Sum of linear fractional functions

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; COMPUTATIONAL GEOMETRY; FUNCTIONS; PROBLEM SOLVING; ROBUSTNESS (CONTROL SYSTEMS);

COMBINATORIAL OPTIMIZATION; MINIMUM RATIO SPANNING TREE (MRST); PARAMETRIC LINEAR PROGRAMMING; SUM OF LINEAR FRACTIONAL FUNCTIONS (SOLF);

LINEAR PROGRAMMING;

EID: 17444376065     PISSN: 13826905     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10878-005-5485-2     Document Type: Article

References (29)

1. E.M. Arkin, Y.-J. Chiang, M. Held, J.S.B. Mitchell, V. Sacristan, S.S. Skiena, and T.-C. Yang, "On minimum-area hulls," Algorithmica, vol. 21, pp. 119-136, 1998.
2. A.I. Barros, Discrete and Fractional Programming Techniques for Location Models, North Holland, 1998.
3. D.A. Bini and G. Fiorentino, "Numerical computation of polynomial roots: MPSolve-version 2.0," FRISCO Report, 1998.
4. G.R. Bitran and T.L. Magnanti, "Duality and sensitivity analysis for fractional programs," Operations Research, vol. 24, pp. 675-699, 1976.
5. T.M. Chan, "Deterministic algorithms for 2-d convex programming and 3-d online linear programming," Journal of Algorithms, vol. 27, pp. 147-166, 1998.
6. R. Chandrasekaran, "Minimum ratio spanning trees," Networks, vol. 7, pp. 335-342, 1977.
7. B. Chazelle, "On the convex layers of a planar set," IEEE Transactions on Information Theory, vol. 31, no. 4, pp. 509-517, 1985.
8. D.Z. Chen, O. Daescu, X. Hu, X. Wu, and J. Xu, "Determining an optimal penetration among weighted regions in two and three dimensions," Journal of Combinatorial Optimization, for a Special Issue on Optimization Problems in Medical Applications, vol. 5, no. 1, pp. 59-79, 2001.
9. L. Craig, J.L. Zhou, and A.L. Tits, "User's guide for CFSQP Version 2.5," Electrical Eng. Dept. and Institute for Systems Research, University of Maryland, TR-94-16r1, 1994.
10. K. Daniels, "The restrict/evaluate/subdivide paradigm for translational containment," in Proceedings of the Fifth MSI Stony Brook Workshop on Computational Geometry, 1995.
11. H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag: New York, 1987.
12. H. Edelsbrunner and L.J. Guibas, "Topologically sweeping an arrangement," Journal of Computer and System Sciences, vol. 38, pp. 165-194, 1989.
13. D. Eppstein, "Dynamic three-dimensional linear programming," in Proceedings of 32nd Annual IEEE Symposium on Foundations of Computer Science, 1991, pp. 488-494.
14. J.E. Falk and S.W. Palocsay, "Optimizing the sum of linear fractional functions," in Collection: Recent Advances in Global Optimization, C.A. Floudas and P.M. Pardalos (Eds.), 1992, pp. 221-258.
15. R.W. Freund and F. Jarre, "Solving the sum-of-ratios problem by an interior-point method," Journal of Global Optimization, vol. 19, no. 1, pp. 83-102, 2001.
16. L. Ingber, Adaptive Simulated Annealing, Optimization Algorithm for Nonlinear and Stochastic Systems, http://www.ingber.com/#ASA-CODE.
17. M. Jelasity and J. Dombi, GAS Genetic Algorithm, ftp://ftp.jate.u-szeged. hu/pub/math/optimization/GAS/.
18. H. Konno, T. Kuno, and Y. Yajima, "Global minimization of a generalized convex multiplicative function," Journal of Global Optimization, vol. 4, pp. 47-62, 1994.
19. G. Liotta, F.P. Preparata, and R. Tamassia, "Robust proximity queries: An illustration of degree-driven algorithm design," SIAM Journal on Computing, vol. 28, no. 3, pp. 864-889, 1999.
20. J. Majhi, R. Janardan, J. Schwerdt, M. Smid, and P. Gupta, "Minimizing support structures and trapped area in two-dimensional layered manufacturing," Technical Report TR-97-058, Dept of Computer Science, University of Minnesota, Dec. 1997a.
21. J. Majhi, R. Janardan, M. Smid, and P. Gupta, "On some geometric optimization problems in layered manufacturing," in Proceedings of the 5th Workshop on Algorithms and Data Structures, vol. 1272, Springer-Verlag, 1997b, pp. 136-149.
22. N. Megiddo, "Combinatorial optimization with rational objective functions," Mathematics of Operations Research, vol. 4, no. 4, pp. 414-424, 1979.
23. N. Megiddo, "Linear programming in linear time when the dimension is fixed," Journal of ACM, vol. 31, pp. 114-127, 1984.
24. V. Milenkovic, K. Daniels, and Z. Li, "Placement and compaction of nonconvex polygons for clothing manufacture," in Proceedings of the 4th Canad. Conf. Comput. Geom., 1992, pp. 236-243.
25. P.P. Preparata and M.I. Shamos, Computational Geometry: An Introduction, Springer-Verlag: New York, 1985.
26. T. Radzik, "Newton's method for fractional combinatorial optimization," in Proceedings of 33rd Annual IEEE Symposium on Foundations of Computer Science, 1992, pp. 659-669.
27. S. Schaible, "Fractional programming," in Handbook of Global Optimization, R. Horst and P.M. Pardalos (Eds.), Kluwer Academic Publishers, 1995, pp. 495-608.
28. C.C. Skiscim and S. Palocsay, "Minimum spanning trees with sums of ratios," Journal of Global Optimization, vol. 19, pp. 103-120, 2001.
29. H. Yamashita, "Efficient algorithms for minimizing the sum and the product of linear fractional functions," Master Thesis, Dept. of Industrial Engineering and Management, Tokyo Institute of Technology, 1997.