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

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

## Abstract

We consider a nonconvex programming problem of minimizing a linear functioncx over a convex setX⊂ℝn 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).

## Zusammenfassung

Wir betrachten eine konvexe Optimierungsaufgabe min {cx:x∈X⊂ℝn,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 ℝp behandelt wird. Der dadurch entstandene Algorithmus ist sehr wirksam für den Fall, daßp (im Verhältnis zun) klein ist.

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## References

1. Floudas, C. A., Hansen, P., Jaumard, B.: Reformulation of two bond portfolio optimization models (to appear in J. Global Optimization).

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).

3. Geoffrion, A.: Solving bicriterion mathematical programs. Oper. Res.15, 39–54 (1967).

4. Henderson, J. M., Quandt, R. E.: Microeconomic theory. New York: McGraw Hill 1971.

5. Horst, R., Tuy, H.: Global optimization: Deterministic approaches. Berlin Heidelberg New York Tokyo: Springer 1990.

6. Horst, R., Thoai, N. V., Tuy, H.: Outer approximation by polyhedral convex sets. Oper. Res. Spektrum9, 153–159 (1987).

7. Horst, R., Thoai, N. V., Tuy, H.: On an outer approximation concept in global optimization. Optimization20, 255–264 (1989).

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).

9. Konno, H., Inori, M.: Bond portfolio optimization by bilinear fractional programming. J. Oper. Res. Soc. Japan32, 143–158 (1988).

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)

11. Kono, H., Kuno, T.: Generalized linear multiplicative and fractional programming. Ann. Oper. Res.25, 147–162 (1990).

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).

13. Muu, L. D.: An algorithm for solving convex programs with an additional convex-concave constraint (to appear in Math. Programming (1993)).

14. Pardalos, P. M.: Polynomial time algorithms for some classes of constrained nonconvex quadratic problems. Preprint, Computer Science Department, The Pennsylvania State University (1988).

15. Thach, P. T., Burkard, R. E., Oettli, W.: Mathematical programs with a two-dimensional reverse convex constraint. J. Global Optimization1, 145–154 (1991).

16. Thoai, N. V.: A modified version of Tuy's method for solving d.c. Programming problems. Optimization9, 665–674 (1988).

17. Thoai, N. V.: A global optimization approach for solving the convex multiplicative programming problem. J Global Optimization1, 341–357 (1991).

18. Tuy, H.: Concave minimization under linear constraints with special structure. Optimization16, 335–352 (1985).

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.

20. Tuy, H.: The complementary convex structure in global optimization. J. Global Optimization2, 21–40 (1992).

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.

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Van Thoai, N. Canonical D. C. programming techniques for solving a convex program with an additional constraint of multiplicative type. Computing 50, 241–253 (1993). https://doi.org/10.1007/BF02243814

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• DOI: https://doi.org/10.1007/BF02243814

• 90C30

### Key words

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