Geometric Programming: Duality in Quadratic Programming and $l_p $-Approximation II (Canonical Programs)

Abstract

The duality theory of geometric programming as developed by Duflin, Peterson and Zener [7] is based on abstract properties shared by certain classical inequalities, such as Cauchy’s arithmetic-geometric mean inequality and Hölder’s inequality. Inequalities with these abstract properties have been termed “geometric inequalities” [7, p. 195]. In this sequence of papers [15], [16], [17] a new geometric inequality is established and used to extend the “refined duality theory” for “posynomial” geometric programs [6] and [7, Chap. VI]. This extended duality theory treats both “quadratically-constrained quadratic programs” and “$l_p $-constrained $l_p $-approximation (regression) problems” through a rather novel and unified formulation of these two classes of programs. This work generalizes some of the work of others on (linearly-constrained) quadratic programs and provides a new explicit formulation of duality for constrained approximation problems. Duality theories have been developed for a large class of programs, namely all convex programs, but those theories (when applied to the programs considered here) are not nearly as strong as the theory developed here. This theory has virtually all of the desirable features [7, p. 11] of its analogue for posynomial programs, and its proof provides useful computational procedures.