, Volume 50, Issue 3, pp 241–253 | Cite as

Canonical D. C. programming techniques for solving a convex program with an additional constraint of multiplicative type

  • N. Van Thoai


We consider a nonconvex programming problem of minimizing a linear functioncx over a convex setX⊂ℝ n with an additional constraint ofmultiplicative type\(\prod _{i = 1}^p \psi _i (x) \leqslant 1\), where the functionsψ i are convex and positive onX. The main idea of our approach is to transform this problem, by usingp additional variables, into acanonical d.c. programming problem with the special structure that thereverse convex constraint involved does only depend on the newly introduced variables. This special structure suggests modifying certain techniques in d.c. programming in a way that the operations handling the nonconvexity are actually performed in the space of the additional variables. The resulting algorithm works very well whenp is small (in comparison withn).

AMS Subject Classification


Key words

Canonical d.c. programming multiplicative programming global optimization outer approximation 

Kanonische d.c. Optimierungstechniken für eine konvexe Optimierungsaufgabe mit einer multiplikativen Nebenbedingung


Wir betrachten eine konvexe Optimierungsaufgabe min {cx:x∈X⊂ℝ n ,X konvex} mit einer zusätzlichen multiplikativen Nebenbedingung der Form\(\prod _{i = 1}^p \psi _i (x) \leqslant 1\), wobei die Funktionenψ i (i=1,...,p) konvex und positiv aufX sind. Mit Hilfe vonp zusätzlichen Variablen transformieren wir diese Aufgabe in eine speziellekanonische d.c. Optimierungsaufgabe, in der die “Reverse convex”—Nebenbedingung nur von den neu eingeführten Variablen abhängt. Dadurch können die Methoden in der kanonischen d.c. Optimierung so ergänzt und modifiziert werden, daß die Nichtkonvexität nur im ℝ p behandelt wird. Der dadurch entstandene Algorithmus ist sehr wirksam für den Fall, daßp (im Verhältnis zun) klein ist.


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  1. [1]
    Floudas, C. A., Hansen, P., Jaumard, B.: Reformulation of two bond portfolio optimization models (to appear in J. Global Optimization).Google Scholar
  2. [2]
    Floudas, C. A., Pardalos, P. M.: A collection of test problems for constrained global optimization algorithms. Berlin Heidelberg New York Tokyo: Springer 1990 (Lecture Notes in Computer Science, 455).Google Scholar
  3. [3]
    Geoffrion, A.: Solving bicriterion mathematical programs. Oper. Res.15, 39–54 (1967).Google Scholar
  4. [4]
    Henderson, J. M., Quandt, R. E.: Microeconomic theory. New York: McGraw Hill 1971.Google Scholar
  5. [5]
    Horst, R., Tuy, H.: Global optimization: Deterministic approaches. Berlin Heidelberg New York Tokyo: Springer 1990.Google Scholar
  6. [6]
    Horst, R., Thoai, N. V., Tuy, H.: Outer approximation by polyhedral convex sets. Oper. Res. Spektrum9, 153–159 (1987).CrossRefGoogle Scholar
  7. [7]
    Horst, R., Thoai, N. V., Tuy, H.: On an outer approximation concept in global optimization. Optimization20, 255–264 (1989).Google Scholar
  8. [8]
    Horst, R., Thoai, N. V., de Vries, J.: On finding new vertices and redundant constraints in cuting plane algorithms for global optimization. Oper. Res. Letters7, 85–90 (1988).CrossRefGoogle Scholar
  9. [9]
    Konno, H., Inori, M.: Bond portfolio optimization by bilinear fractional programming. J. Oper. Res. Soc. Japan32, 143–158 (1988).MathSciNetGoogle Scholar
  10. [10]
    Konno, H., Kuno, T.: Linear multiplicative programming. IHSS 89-13, Institute of Human and Social Sciences, Tokyo Institute of Technology (1989). (Forthcoming in Math. Programming)Google Scholar
  11. [11]
    Kono, H., Kuno, T.: Generalized linear multiplicative and fractional programming. Ann. Oper. Res.25, 147–162 (1990).Google Scholar
  12. [12]
    Kuno, T., Konno H., Yamamoto, Y.: A parametric successive underestimation method for convex programming problems with an additional convex multiplicative constraint. IHSS 90-23, Institute of Human and Social Sciences, Tokyo Institute of Technology (1990).Google Scholar
  13. [13]
    Muu, L. D.: An algorithm for solving convex programs with an additional convex-concave constraint (to appear in Math. Programming (1993)).Google Scholar
  14. [14]
    Pardalos, P. M.: Polynomial time algorithms for some classes of constrained nonconvex quadratic problems. Preprint, Computer Science Department, The Pennsylvania State University (1988).Google Scholar
  15. [15]
    Thach, P. T., Burkard, R. E., Oettli, W.: Mathematical programs with a two-dimensional reverse convex constraint. J. Global Optimization1, 145–154 (1991).CrossRefGoogle Scholar
  16. [16]
    Thoai, N. V.: A modified version of Tuy's method for solving d.c. Programming problems. Optimization9, 665–674 (1988).Google Scholar
  17. [17]
    Thoai, N. V.: A global optimization approach for solving the convex multiplicative programming problem. J Global Optimization1, 341–357 (1991).CrossRefGoogle Scholar
  18. [18]
    Tuy, H.: Concave minimization under linear constraints with special structure. Optimization16, 335–352 (1985).Google Scholar
  19. [19]
    Tuy, H.: A general deterministic approach to global optimization via d.c. programming. In: Hiriart-Urruty, J. B. (ed.), Fermat Days 1985: Mathematics for Optimization, Amsterdam Elsevier: pp. 137–162, 1986.Google Scholar
  20. [20]
    Tuy, H.: The complementary convex structure in global optimization. J. Global Optimization2, 21–40 (1992).CrossRefGoogle Scholar
  21. [21]
    Yajima, Y., Kuno, T., Konno, H., Yamamoto, Y.: Convex programs with an additional constraint on the product of several convex functions. Presented at 14th International Symposium on Mathematical Programming, Amsterdam 5–9, 1991.Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • N. Van Thoai
    • 1
  1. 1.Fachbereich IVMathematik Universität TrierTrierGermany

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