An outcome space branch and bound-outer approximation algorithm for convex multiplicative programming
The article under review is a valuable contribution to both theory and, in particular, numerical analysis of so-called convex multiplicative programming. By definition, in this field of nonlinear minimization the objective function is a finite product of convex functions over some compact convex set \(D\). Such a problem \((P_D)\) arises in economic analysis, portfolio and multi-objective optimization, and in VLSI design. In order to solve this NP-hard problem with computational savings, the author bases his development of an approximate algorithm on an equivalent quasiconcave minimization problem, called outcome space problem \((P_Y)\). Assuming \(D\) finitely convex and differentiably defined, the algorithm firstly implies outer approximation of the nonempty, compact outcome space \(Y\) by decreasing polytopes, found by a nonlinear convex program, resolution of a system of linear (in)equalities, and a ``cut off'' from one iteration to the other. Herewith, the algorithm works ``economized''. Secondly, we find branching and, thirdly, bounding techniques from linear programming, prepared by suitable notions and notations. The author presents an (upon termination) finite convergence theorem to an \(\varepsilon\)-optimal solution, and gives detailed computational issues. Finally, this carefully written investigation in global optimization becomes applied to a numerical example.
scientific article; zbMATH DE number 1455284