The proximity of (algebraic) geometric programming to linear programming

Geometric programming with (posy)monomials is known to be synonomous with linear programming, Ifais note reduces algebraic programming to geometric programming with (posy)binomials. Carnegie-Mellon University, Pittsburgh, Pennsylvania, 15213. Partially supported by the Army under research grant DA--AROD-31-124-71-G17. Northwestern University, Evanston, Illinois, 60201. Partially supported by the Northwestern University Urban Systems Engineering Center under research grant #731. 1. introductiorio In [3] we demonstrate the reduction of each well-posed "algebraic program" to an equivalent "posynomial program" in which a posynomial is to be minimized subject only to inequality posynomial constraints (some of which may have a "reversed direction"). Of course, each of those posynomial programs can be reformulated so that its objective function is a (posy)monomial in that it includes only one posynomial term. (To make such a reformulation, simply minimize an additional independent variable that is constrained to be at least as large as the posynomial objective function.) The purpose of this note is to show that each of those posynomial programs can be further reformulated so that every constraint function is a (posy)binomial in that it includes at most only two posynomial terms. This reformulation is rather striking in view of Federowicz's observation [5, Appendix D] that (posy)monomial programming (i.e., posynomial programming with (posy)monomial objective and constraint functions) is synonomous with linear programming. In fact, we suspect that the resulting proximity of algebraic programming to linear programming may have important computational and theoretical implications. 2. The Reformulation, In [3] (and [2]) we actually show that one need only consider "prototype posynomial constraints'