Journal of Global Optimization

, Volume 20, Issue 2, pp 133–154 | Cite as

Semidefinite Relaxations of Fractional Programs via Novel Convexification Techniques

  • Mohit Tawarmalani
  • Nikolaos V. Sahinidis
Article

Abstract

In a recent work, we introduced the concept of convex extensions for lower semi-continuous functions and studied their properties. In this work, we present new techniques for constructing convex and concave envelopes of nonlinear functions using the theory of convex extensions. In particular, we develop the convex envelope and concave envelope of z=x/y over a hypercube. We show that the convex envelope is strictly tighter than previously known convex underestimators of x/y. We then propose a new relaxation technique for fractional programs which includes the derived envelopes. The resulting relaxation is shown to be a semidefinite program. Finally, we derive the convex envelope for a class of functions of the type f(x,y) over a hypercube under the assumption that f is concave in x and convex in y.

Convex Envelopes Convex Extensions Fractional Programs Semidefinite Relaxations Disjunctive Programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Al-Khayyal, F.A. and Falk, J.E. (1983), Jointly Constrained Biconvex Programming. Mathematics of Operations Research 8: 273-286.Google Scholar
  2. 2.
    Alizadeh, F. (1995), Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization. SIAM Journal of Optimization 5: 13-51.Google Scholar
  3. 3.
    Barros, A.I. (1998), Discrete and Fractional Programming Technqiues for Location Models. Kluwer Academic Publishers, Dordrecht.Google Scholar
  4. 4.
    Bradley, S.P. and Frey, C.S. (1974), Fractional Programming with Homogenous Functions. Operations Research 22: 350-357.Google Scholar
  5. 5.
    Cambini, A., Castagnoli, E., Martein, L., Mazzoleni, P. and Schaible, S., (eds.) (1990), Generalized Convexity and Fractional Programming with Economic Applications, volume 345 of Lecture Notes in Economics and Mathematical Systems. Springer Verlag.Google Scholar
  6. 6.
    Falk, J.E. and Polocsay, S.W. (1994), Image Space Analysis of Generalized Fractional Programs. Journal of Global Optimization 4: 63-88.Google Scholar
  7. 7.
    Fujie, T. and Kojima, M. (1997), Semidefinite Programming Relaxation for Nonconvex Quadratic Programs. Journal of Global Optimization 10: 367-380.Google Scholar
  8. 8.
    Gill, P.E., Murray, W. and Saunders, M.A. (1999), User's Guide for SNOPT 5.3: A FORTRAN Package for Large-Scale Nonlinear Programming. Technical report, Technical Report, University of California, San Diego and Stanford University, CA.Google Scholar
  9. 9.
    Kojima, M. and Tunçel, L. (1999), Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets. Technical report, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Japan.Google Scholar
  10. 10.
    Konno, H. and Kuno, T. (1990), Generalized Linear Multiplicative and Fractional Programming. Annals of Operations Research 25: 147-162.Google Scholar
  11. 11.
    Li, H. (1994), A Global Approach for General 0.1 Fractional Programming. European Journal of Operational Research 73: 590-596.Google Scholar
  12. 12.
    Lobo, M.S., Vandenberghe, L., Boyd, S. and Lebret, H. (1998), Applications of Second-Order Cone Programming. Linear Algebra Applications 284: 193-228.Google Scholar
  13. 13.
    Lovàsz, L. and Schrijver, A. (1991), Cones of Matrices and Set-functions and 0-1 Optimization.SIAM Journal on Optimization 1: 166-190.Google Scholar
  14. 14.
    Matsui, T. (1996), NP-Hardness of Linear Multiplicative Programming and Related Problems. Journal of Global Optimization 9: 113-119.Google Scholar
  15. 15.
    McCormick, G.P. (1976), Computability of Global Solutions to Factorable Nonconvex Programs: Part I-Convex Underestimating Problems. Mathematical Programming 10: 147-175.Google Scholar
  16. 16.
    McCormick, G.P. (1982), Nonlinear Programming: Theory, Algorithms and Applications. John Wiley and Sons.Google Scholar
  17. 17.
    Meister, B. and Oettli, W. (1967), On the Capacity of a Discrete, Constant Channel. Information and Control 11: 341-351.Google Scholar
  18. 18.
    Murtagh, B.A. and Saunders, M.A. (1995), MINOS 5.5 User's Guide. Technical report, Technical Report SOL 83-20R, Systems Optimization Laboratory, Department of Operations Research, Stanford University, CA.Google Scholar
  19. 19.
    Nesterov, Y. and Nemirovskii, A. (1994), Interior-Point Polynomial Algorithms in Convex Programming, volume 13. SIAM Studies in Applied Mathematics, Philadelphia.Google Scholar
  20. 20.
    Poljak, S., Rendl, F. and Wolkowicz, H. (1994), A Recipe for Semidefinite Relaxations of (0,1) Quadratic Programming. Technical report, University of Waterloo.Google Scholar
  21. 21.
    Quesada, I. and Grossmann, I.E. (1995), A Global Optimization Algorithm for Linear Fractional and Bilinear Programs. Journal of Global Optimization 6: 39-76.Google Scholar
  22. 22.
    Rikun, A.D. (1997), A Convex Envelope Formula for Multilinear Functions. Journal of Global Optimization 10: 425-437.Google Scholar
  23. 23.
    Rockafellar, R.T. (1970), Convex Analysis. Princeton Mathematical Series. Princeton University Press.Google Scholar
  24. 24.
    Schaible, S. (1995), Fractional programming with sums of ratios. In: E. Castagnoli and J. Giorgi (eds.), Proceedings of the Workshop held in Milan on March 28, 1995, pp. 163-175.Google Scholar
  25. 25.
    Schaible, S. (1996), Fractional Programming. In S.I. Gass and C.M. Harris (eds.), Encyclopedia of Operations Research and Management Science, Boston: Kluwer Academic Publishers, pp 234-237.Google Scholar
  26. 26.
    Sherali, H.D. (1997), Convex Envelopes of Multilinear Functions over a Unit Hypercube and over Special Discrete Sets. Acta Mathematica Vietnamica 22: 245-270.Google Scholar
  27. 27.
    Stancu-Minasian, I.M. (1990), On Some Fractional Programming Models Occurring in Minimum-Risk Problems. In Cambini et al. [5], pp. 295-324.Google Scholar
  28. 28.
    Stancu-Minasian, I.M. (1997), Fractional Programming. Netherlands: Kluwer Academic Publishers.Google Scholar
  29. 29.
    Tawarmalani, M., Ahmed, S. and Sahinidis, N.V. (1999), Global Optimization of 0.1 Hyperbolic Programs. Journal of Global Optimization. http://archimedes.scs.uiuc.edu/papers/fractional.pdf.Google Scholar
  30. 30.
    Tawarmalani, M. and Sahinidis, N.V. (1999), Global Optimization of Mixed Integer Nonlinear Programs: A Theoretical and Computational Study. Mathematical Programming. http://archimedes.scs.uiuc.edu/papers/comp.pdf.Google Scholar
  31. 31.
    Tawarmalani, M. and Sahinidis, N.V. (2000), Convex Extensions and Convex Envelopes of l.s.c. Functions. Mathematical Programming. http://archimedes.scs.uiuc.edu/papers/extensions.pdf.Google Scholar
  32. 32.
    Vandenberghe, L. and Boyd, S. (1996), Semidefinite Programming. SIAM Review 38: 49-95.Google Scholar
  33. 33.
    Yee, T.F. and Grossmann, I.E. (1990), Simultaneous Optimization Models for Heat Integration-II. Heat Exchanger Network Synthesis. Computers and Chemical Engineering 14: 1165-1184.Google Scholar
  34. 34.
    Zamora, J.M. and Grossmann, I.E. (1998), MINLP Model for Heat Exchanger Networks. Computers & Chemical Engineering 22: 367-384.Google Scholar
  35. 35.
    Zamora, J.M. and Grossmann, I.E. (1999), A Branch and Contract Algorithm for problems with Concave Univariate, Bilinear and Linear Fractional Terms. Journal of Global Optimization 14: 217-249.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Mohit Tawarmalani
    • 1
  • Nikolaos V. Sahinidis
    • 2
  1. 1.Department of Mechanical & Industrial EngineeringThe University of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Chemical EngineeringThe University of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations