# DUALITY THEORY FOR COMPOSITE GEOMETRIC PROGRAMMING

Wang, Ya-Ping (2013) DUALITY THEORY FOR COMPOSITE GEOMETRIC PROGRAMMING. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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## Abstract

This research develops an alternative approach to the duality theory of the well-known subject of Geometric Programming (GP), and uses this to then develop a duality theory for Quadratic Geometric Programming (QGP), which is an extension of GP; it then develops a duality theory for (convex) Composite Geometric Programming (CGP), which in turn, is a generalization of QGP. The building block of GP is a special class of functions called posynomials, which are summations of terms, where the logarithm of each term is a linear function of the logarithms of its design variables. Such log-linear relationships often appear as an empirical fit in numerous engineering applications, most notably in engineering design. At other times, these relationships may simply follow the dictates of the laws of nature and/or economics (Varian 1978). Many functions that describe engineering systems are posynomials and hence GP is especially suitable for handling optimization problems involving such functions. For instance, the so-called Machining Economics Problem (MEP) for conventional metals can be handled successfully by GP (Tsai, 1986). The defining relationship of the tool life as a function of machining variables such as cutting speed, depth of cut, feed rate, tool change downtime, etc., is typically log-linear.
However, when the logarithm of the tool life for a given alloy is a quadratic (instead of a linear) function of the logarithms of the machining variables, GP is not able to handle this kind of MEP (Hough, 1978; Hough and Goforth, 1981a, b, c). Jefferson and Scott (1985) introduced QGP to successfully handle the quadratic version of the MEP. They also gave primal-dual formulations of the QGP problem, an optimality condition, and three illustrative numerical examples of MEP (Jefferson and Scott 1985, p.144). A strong duality theorem for QGP was later proved by Fang and Rajasekera (1987) using a dual perturbation approach and two simple geometric inequalities.
However, more detailed duality theory for QGP and for CGP comparable to the development for GP by Duffin, Peterson, and Zener in chapters IV and VI of their seminal text (1967) are yet to be developed. This theory is rooted in the duality principle for ‘conjugate’ pairs of convex functions, which translates the analysis of one optimization problem into an equivalent, yet very different optimization problem. It allows us to view the problem from two different angles, often with new insights and with other remarkable consequences such as suggesting an easier solution approach. In this thesis, we extend QGP problems to the more general CGP problems to account for the even more general log-convex (as opposed to log-linear) relationships. The aim of this dissertation is to develop a comprehensive duality theory for both QGP and CGP.

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## Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Wang, Ya-Pingyaw13@pitt.eduYAW13
ETD Committee:
Committee ChairRajgopal, Jayantrajgopal@pitt.eduRAJGOPAL
Committee MemberProkopyev, Olegdroleg@pitt.eduDROLEG
Committee MemberSaure, Denisdsaure@pitt.eduDSAURE
Committee MemberMao, Zhi-Hongzhm4@pitt.eduZHM4
Date: 28 June 2013
Date Type: Publication
Defense Date: 14 December 2012
Approval Date: 28 June 2013
Submission Date: 11 March 2013
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 164
Institution: University of Pittsburgh
Schools and Programs: Swanson School of Engineering > Industrial Engineering
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Geometric Programming, Optimization, Duality
Date Deposited: 28 Jun 2013 20:02
URI: http://d-scholarship.pitt.edu/id/eprint/17746