STRONG DUALITY IN GEOMETRIC OPTIMIZATION USING A CONIC FORMULATION

Geometric optimization is an important class of problems that has many applications,
especially in engineering design.

In this talk, we provide new simplified proofs for the well-known associated duality
theory, using conic optimization. After introducing suitable convex cones and
studying their properties, we model geometric optimization problems with a conic
formulation, which allows us to apply the powerful duality theory of conic
optimization and derive the duality results known for geometric optimization.

Keywords: geometric optimization, duality theory, conic optimization.

INTRODUCTION

Geometric optimization forms an important class of problems that enables
practitioners to model a large variety of real-world applications, mostly in the
field of engineering design.

Although not convex itself, a geometric optimization problem can be easily
transformed into a convex problem, for which a Lagrangean dual can be explicitly
written. Several duality results are known for this pair of problems, some being
mere consequences of convexity (e.g. weak duality), others being specific to this
particular class of problems (e.g. the absence of a duality gap).

These properties were first studied in the late sixties, and can be found for
example in the book of Zener, Duffin and Peterson [1]. The aim of this talk is to
derive these results using the machinery of duality for conic optimization, which
has in our opinion the advantage of simplifying the proofs.

In order to use this setting, we start by defining an appropriate convex cone that
allows us to express geometric optimization problems as conic programs. The first
step we take consists in studying some properties of this cone (e.g. closedness) and
determine its dual. We are then in position to apply the general duality theory for
conic optimization [2] to our problems and find in a rather seamless way the various
well-known duality theorems of geometric optimization.

REFERENCES

[1] C. Zener, R. J. Duffin and E. L. Peterson, Geometric programming, John Wiley &
Sons, New York, 1967.

[2] J. F. Sturm, Primal-dual interior-point approach to semidefinite programming,
Ph.D. thesis, Erasmus Universiteit Rotterdam, The Netherlands, 1997.