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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


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).


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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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).

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AMS Subject Classification

  • 90C30

Key words

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