A global optimization approach for solving the convex multiplicative programming problem.(English)Zbl 0752.90056

Summary: We consider a convex multiplicative programming problem of the form $$\min\{f_ 1(x)\cdot f_ 2(x): x\in X\}$$, where $$X$$ is a compact convex subset of $$\mathbb{R}^ n$$ and $$f_ 1$$, $$f_ 2$$ 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 $$\mathbb{R}^ 2_ +$$. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm terminates when the function $$f_ i$$, $$(i=1,2)$$ are affine-linear and $$X$$ is a polytope and it is convergent for the general convex case.

MSC:

 90C25 Convex programming 90-08 Computational methods for problems pertaining to operations research and mathematical programming
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References:

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