# A tutorial on geometric programming

• Stephen Boyd
• Seung-Jean Kim
• Lieven Vandenberghe
• Arash Hassibi
Educational Section

## Abstract

A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even large-scale GPs extremely efficiently and reliably; at the same time a number of practical problems, particularly in circuit design, have been found to be equivalent to (or well approximated by) GPs. Putting these two together, we get effective solutions for the practical problems. The basic approach in GP modeling is to attempt to express a practical problem, such as an engineering analysis or design problem, in GP format. In the best case, this formulation is exact; when this is not possible, we settle for an approximate formulation. This tutorial paper collects together in one place the basic background material needed to do GP modeling. We start with the basic definitions and facts, and some methods used to transform problems into GP format. We show how to recognize functions and problems compatible with GP, and how to approximate functions or data in a form compatible with GP (when this is possible). We give some simple and representative examples, and also describe some common extensions of GP, along with methods for solving (or approximately solving) them.

## Keywords

Convex optimization Geometric programming Generalized geometric programming Interior-point methods

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## Authors and Affiliations

• Stephen Boyd
• 1
• Seung-Jean Kim
• 1
• Lieven Vandenberghe
• 2
• Arash Hassibi
• 3
1. 1.Information Systems Laboratory, Department of Electrical EngineeringStanford UniversityStanfordUSA
2. 2.Department of Electrical EngineeringUniversity of CaliforniaLos AngelesUSA
3. 3.Clear Shape Technologies, Inc.SunnyvaleUSA