Advertisement

Mathematical Programming

, Volume 141, Issue 1–2, pp 103–120 | Cite as

An FPTAS for optimizing a class of low-rank functions over a polytope

  • Shashi MittalEmail author
  • Andreas S. Schulz
Full Length Paper Series A

Abstract

We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of non-linear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Pareto-optimal front of the linear functions which constitute the given low-rank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity on the objective function. For the special case of quasi-concave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with non-linear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest.

Keywords

Non-convex optimization Combinatorial optimization Approximation schemes 

Mathematics Subject Classification (2000)

90C20 90C26 90C27 90C29 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Al-Khayyal F.A., Falk J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8, 273–286 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atamtürk A., Narayanan V.: Polymatroids and mean-risk minimization in discrete optimization. Oper. Res. Lett. 36, 618–622 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benson H.P., Boger G.M.: Multiplicative programming problems: analysis and efficient point search heuristic. J. Optim. Theory App. 94, 487–510 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bertsekas D., Nedić A., Ozdaglar A.: Convex Analysis and Optimization. Athena Scientific, Belmont, MA (2003)zbMATHGoogle Scholar
  5. 5.
    Diakonikolas, I., Yannakakis, M.: Succinct approximate convex pareto curves. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 74–83. San Francisco, CA (2008)Google Scholar
  6. 6.
    Eisenberg E.: Aggregation of utility functions. Manag. Sci 7, 337–350 (1961)CrossRefGoogle Scholar
  7. 7.
    Falk J.E., Palocsay S.W.: Optimizing the sum of linear fractional functions. In: Floudas, C.A., Pardalos, P.M. (eds) Recent Advances in Global Optimization, pp. 228–258. Kluwer, Dordrecht (1992)Google Scholar
  8. 8.
    Feige, U., Mirrokni, V.S., Vondrák, J.: Maximizing non-monotone submodular functions. In: 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 461–471. Providence, RI (2007)Google Scholar
  9. 9.
    Goyal, V., Ravi, R.: An FPTAS for minimizing a class of quasi-concave functions over a convex domain. Tech. rep., Tepper School of Business, Carnegie Mellon University (2009)Google Scholar
  10. 10.
    Goyal V., Genc-Kaya L., Ravi R.: An FPTAS for minimizing the product of two non-negative linear cost functions. Math. Program. 126, 401–405 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grötschel M., Lovász L., Schrijver A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  12. 12.
    Håstad J.: Some optimal inapproximability results. J. Assoc. Comput. Mach. 48, 798–859 (2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    Horst, R., Pardalos, P.M. (eds): Handbook of Global Optimization, vol. 1. Kluwer, Dordrecht, The Netherlands (1995)Google Scholar
  14. 14.
    Kelner, J.A., Nikolova, E.: On the hardness and smoothed complexity of quasi-concave minimization. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 472–482. Providence, RI (2007)Google Scholar
  15. 15.
    Kern W., Woeginger G.J.: Quadratic programming and combinatorial minimum weight product problems. Math. Program. 110, 641–649 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Konno H.: A cutting plane algorithm for solving bilinear programs. Math. Program. 11, 14–27 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Konno H., Kuno T.: Linear multiplicative programming. Math. Program. 56, 51–64 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Konno H., Thach T., Tuy H.: Optimization on Low Rank Nonconvex Structures. Kluwer, Dordrecht, The Netherlands (1996)Google Scholar
  19. 19.
    Konno H., Gao C., Saitoh I.: Cutting plane/tabu search algorithms for low rank concave quadratic programming problems. J. Glob. Optim. 13, 225–240 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kuno T.: Polynomial algorithms for a class of minimum rank-two cost path problems. J. Glob. Optim. 15, 405–417 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lovász L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds) Mathematical Programming—The State of the Art, pp. 235–257. Springer, Berlin (1983)CrossRefGoogle Scholar
  22. 22.
    Matsui T.: NP-hardness of linear multiplicative programming and related problems. J. Glob. Optim. 9, 113–119 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mittal, S., Schulz, A.S.: A general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one. In: APPROX-RANDOM, Cambridge, MA, Lecture Notes in Computer Science, vol. 5171, pp. 179–192 (2008)Google Scholar
  24. 24.
    Nesterov Y., Nemirovskii A.: Interior Point Polynomial Methods in Convex Programming. Society for Industrial and Applied Mathematics, Philadelphia, PA (1961)Google Scholar
  25. 25.
    Nikolova, E.: Approximation algorithms for reliable stochastic combinatorial optimization. In: APPROX-RANDOM, Barcelona, Spain, Lecture Notes in Computer Science, vol. 6302, pp. 338–351 (2010)Google Scholar
  26. 26.
    Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, pp. 86–92, Redondo Beach, CA (2000)Google Scholar
  27. 27.
    Pardalos P.M., Vavasis S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 1, 15–22 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Porembski M.: Cutting planes for low-rank like concave minimization problems. Oper. Res. 52, 942–953 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Safer, H.M., Orlin, J.B.: Fast approximation schemes for multi-criteria combinatorial optimization. Tech. rep., Operations Research Center, Massachusetts Institute of Technology (1995)Google Scholar
  30. 30.
    Safer, H.M., Orlin, J.B.: Fast approximation schemes for multi-criteria flow, knapsack, and scheduling problems. Tech. rep., Operations Research Center, Massachusetts Institute of Technology (1995)Google Scholar
  31. 31.
    Safer, H.M., Orlin, J.B., Dror, M.: Fully polynomial approximation in multi-criteria combinatorial. Tech. rep., Operations Research Center, Massachusetts Institute of Technology (2004)Google Scholar
  32. 32.
    Schaible S.: A note on the sum of a linear and linear-fractional function. Nav. Res. Logist. 24, 691–693 (1977)CrossRefzbMATHGoogle Scholar
  33. 33.
    Schaible S., Shi J.: Fractional programming: the sum-of-ratios case. Optim. Methods Softw. 18, 219–229 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sherali H.D., Alameddine A.: A new reformulation-linearization technique for bilinear programming problems. J. Glob. Optim. 2, 379–410 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Vavasis S.A.: Approximation algorithm for indefinite quadratic programming. Math. Program. 57, 279–311 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Amazon.comSeattleUSA
  2. 2.Sloan School of Management and Operations Research CenterMassachusetts Institute of Technology, E62-569CambridgeUSA

Personalised recommendations