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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
Article

Abstract

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 

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Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

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

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