Monotonicity in the Framework of Generalized Convexity

  • Hoang Tuy
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 77)


An increasing function f : R n R ∪ +∞ is a function such that f(x′) f(x) whenever x′ x (component-wise). A downward set G R n is a set such that x G whenever x′ x for some x′ G. We present a geometric theory of monotonicity in which increasing functions relate to downward sets in the same way as convex functions relate to convex sets. By giving a central role to a separation property of downward sets similar to that of convex sets, a theory of monotonic optimization can be developed which parallels d.c. optimization in several respects.


Monotonicity Downward sets Normal sets Separation property Polyblock Increasing functions Monotonic functions Difference of monotonic functions (d.m. functions) Abstract convex analysis Global optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. E.F. Beckenbach and R. Bellman, Inequalities, Springer-Verlag 1961.Google Scholar
  2. A. Ben-Tal and A. Ben-Israel, F-convex functions: Properties and applications, in: Generalized concavity in optimization and economics, eds. S. Schaible and W.T. Ziemba, Academic Press, New York 1981.Google Scholar
  3. Z. First, S.T. Hackman and U. Passy, Local-global properties of bifunctions, Journal of Optimization Theory and Applications 73 (1992) 279–297.MathSciNetCrossRefzbMATHGoogle Scholar
  4. S.T. Hackman and U. Passy, Projectively-convex sets and functions, Journal of Mathematical Economics 17 (1988) 55–68.MathSciNetCrossRefzbMATHGoogle Scholar
  5. N.T. Hoai Phuong and H. Tuy, A Monotonicity Based Approach to Nonconvex Quadratic Minimization, Vietnam Journal of Mathematics 30:4 (2002) 373–393.MathSciNetzbMATHGoogle Scholar
  6. N.T. Hoai Phuong and H. Tuy, A unified approach to generalized fractional programming, Journal of Global Optimization, 26 (2003) 229–259.CrossRefzbMATHGoogle Scholar
  7. R. Horst and H. Tuy, Global Optimization (Deterministic Approaches), third edition, Springer-Verlag, 1996.Google Scholar
  8. H. Konno and T. Kuno, Generalized multiplicative and fractional programming, Annals of Operations Research, 25 (1990) 147–162.MathSciNetCrossRefzbMATHGoogle Scholar
  9. H. Konno, Y. Yajima and T. Matsui, Parametric simplex algorithms for solving a special class of nonconvex minimization problems, Journal of Global Optimization, 1 (1991) 65–81.MathSciNetCrossRefzbMATHGoogle Scholar
  10. H. Konno, P.T. Thach and H. Tuy, Optimization on Low Rank Nonconvex Structures, Kluwer Academic Publishers, 1997.Google Scholar
  11. D.T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems 319, Springer-Verlag, 1989.Google Scholar
  12. V.L. Makarov and A.M. Rubinov, Mathematical Theory of Economic Dynamic and Equilibria, Springer-Verlag, 1977.Google Scholar
  13. J.-E. Martinez-Legaz, A.M. Rubinov and I. Singer, Downward sets and their separation and approximation properties, Journal of Global Optimization, 23 (2002) 111–137.MathSciNetCrossRefzbMATHGoogle Scholar
  14. P. Papalambros and H.L. Li, Notes on the operational utility of monotonicity in optimization, ASME Journal of Mechanisms, Transmissions, and Automation in Design, 105 (1983) 174–180.CrossRefGoogle Scholar
  15. P. Papalambros and D.J. Wilde, Principles of Optimal Design-Modeling and Computation, Cambridge University Press, 1986Google Scholar
  16. U. Passy, Global solutions of mathematical programs with intrinsically concave functions, in M. Avriel (ed.), Advances in Geometric Programming, Plenum Press, 1980.Google Scholar
  17. A. Rubinov, Abstract Convexity and Global Optimization Kluwer Academic Publishers, 2000.Google Scholar
  18. A. Rubinov, H. Tuy and H. Mays, Algorithm for a monotonic global optimization problem, Optimization, 49 (2001), 205–221.MathSciNetzbMATHGoogle Scholar
  19. I. Singer, Abstract convex analysis, Wiley-Interscience Publication, New York, 1997.zbMATHGoogle Scholar
  20. A. N. Tikhonov, On a reciprocity principle, Soviet Mathematics Doklady, vol.22, pp. 100–103, 1980.zbMATHGoogle Scholar
  21. H. Tuy, Convex programs with an additional reverse convex constraint, Journal of Optimization Theory and Applications 52 (1987) 463–486zbMATHMathSciNetCrossRefGoogle Scholar
  22. H. Tuy, D.C. Optimization: Theory, Methods and Algorithms, in R. Horst and P.M. Pardalos (eds.), Handbook on Global Optimization, Kluwer Academic Publishers, 1995, pp. 149–216.Google Scholar
  23. H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, 1998.Google Scholar
  24. H. Tuy, Normal sets, polyblocks and monotonic optimization, Vietnam Journal of Mathematics 27:4 (1999) 277–300.zbMATHMathSciNetGoogle Scholar
  25. H. Tuy, Monotonic optimization: Problems and solution approaches, SIAM J. Optimization 11:2 (2000), 464–494.zbMATHMathSciNetCrossRefGoogle Scholar
  26. H. Tuy and Le Tu Luc, A new approach to optimization under monotonic constraint, Journal of Global Optimization, 18 (2000) 1–15.MathSciNetCrossRefzbMATHGoogle Scholar
  27. H. Tuy and F. Al-Khayyal, Monotonic Optimization revisited, Preprint, Institute of Mathematics, Hanoi, 2003.Google Scholar
  28. H. Tuy, On global optimality conditions and cutting plane algorithms, Journal of Optimization Theory and Applications, Vol. 118 (2003), No. 1, 201–216.zbMATHMathSciNetCrossRefGoogle Scholar
  29. H. Tuy, M. Minoux and N.T. Hoai-Phuong: Discrete monotonic optimization with application to a discrete location problem, Preprint, Institute of Mathematics, Hanoi, 2004.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Hoang Tuy
    • 1
  1. 1.Institute of MathematicsVietnam

Personalised recommendations