Journal of Global Optimization

, Volume 1, Issue 4, pp 341–357 | Cite as

A global optimization approach for solving the convex multiplicative programming problem

  • Nguyen Van Thoai


We consider a convex multiplicative programming problem of the form

% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \]

where X is a compact convex set of ℝ n and f1, f2 are convex functions which have nonnegative values over X.

Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ℝ22. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions fi, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.

Key words

Multiplicative programming global optimization decomposition outer approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aneja Y. P., Aggarwal V. and Nair K. (1984), On a Class of Quadratic Programming, European Journal of Oper. Res. 18, 62–70.Google Scholar
  2. Bector C. R. and Dahl M. (1974), Simplex Type Finite Iteration Technique and Reality for a Special Type of Pseudo-Concave Quadratic Functions, Cahiers du Centre d'Etudes de Recherche Operationnelle 16, 207–222.Google Scholar
  3. Gabasov, R. and Kirillova, F. M. (1980), Linear Programming Methods, Part 3 (Special Problems), Minsk (in Russian).Google Scholar
  4. Hoffman K. L. (1981), A Method for Globally Minimizing Concave Functions over Convex Sets, Mathematical Programming 20, 22–32.Google Scholar
  5. Horst R. and Thoai N. V. (1989), Modification, Implementation and Comparison of Three Algorithms for Globally Solving Linearly Constrained Concave Minimization Problems, Computing 42, 271–289.Google Scholar
  6. Horst, R. and Tuy, H. (1990), Global Optimization: Deterministic Approaches, Springer-Verlag.Google Scholar
  7. Horst R., Thoai N. V. and Tuy H. (1987), Outer Approximation by Polyhedral Convex Sets, Oper. Res. Spektrum 9, 153–159.Google Scholar
  8. Horst R., Thoai N. V., and Tuy H. (1989), On an Outer Approximation Concept in Global Optimization, Optimization 20, 255–264.Google Scholar
  9. Horst R., Thoai N. V., and de Vries J. (1988), On Finding New Vertices and Redundant Constraints in Cutting Plane Algorithms for Global Optimization, Oper. Res. Letters 7, 85–90.Google Scholar
  10. Konno, H. and Kuno, T. (1989), Linear Multiplicative Programming, IHSS 89-13, Institute of Human and Social Sciences, Tokyo Institute of Technology.Google Scholar
  11. Kuno, T. and Konno, H. (1990), A Parametric Successive Underestimation Method for Convex Multiplicative Programming Problems, IHSS 90-19, Institute of Human and Social Sciences, Tokyo Institute of Technology.Google Scholar
  12. Pardalos, P. M. (1988), Polynomial Time Algorithms for Some Classes of Constrained Nonconvex Quadratic Problems, Preprint, Computer Science Department, The Pennsylvania State University.Google Scholar
  13. Rockafellar R. T. (1970), Convex Analysis, Princeton University Press, Princeton, N.J.Google Scholar
  14. Schaible S. (1976), Minimization of Ratios, J. Optimization Theory Appl. 19, 347–352.Google Scholar
  15. Swarup K. (1966), Programming with Indefinite Quadratic Function with Linear Constraints, Cahiers de Centre d'Etudes de Recherche Operationnelle 8, 133–136.Google Scholar
  16. Tuy H. (1985), Concave Minimization under Linear Constraints with Special Structure, Optimization 16, 335–352.Google Scholar
  17. Tuy H. (1990), The Complementary Convex Structure in Global Optimization, Preprint, Institute of Mathematics, Hanoi.Google Scholar
  18. Tuy H. and Tam B. T. (1990), An Efficient Solution Method for Rank Two Quasiconcave Minimization Problems, Preprint, Institute of Mathematics, Hanoi.Google Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Nguyen Van Thoai
    • 1
  1. 1.Fachbereich IV-MathematikUniversität TrierTrierGermany

Personalised recommendations