Geometric optimization is an important class of problems that has many applications, especially in engineering design. In this article, 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 valid for geometric optimization.

The author has recently proposed a new way of formulating two classical classes of structured convex problems, geometric and l p -norm optimization, using dedicated convex cones [Gli99, GT00]. This approach has some advantages over the traditional formulation: it simplifies the proofs of the well-known associated duality properties (i.e. weak and strong duality) and the design of a polynomial algorithm becomes straightforward. These new proofs rely on the general duality theory valid for convex problems expressed in conic form [SW70, Stu00] and the work on polynomial interior-point methods by Nesterov and Nemirovsky [NN94]. In this paper, we make a step towards the description of a common framework that would include these two classes of problems. Indeed, we introduce an extended variant of the cone for geometric optimization used in [Gli99] and show it is equally suitable to formulate this class of problems. This new cone has the additional advantage of being very similar to the ...

In this article, we demonstrate how to approximate geometric optimization with l p - norm optimization. These two categories of problems are well known in structured convex optimization. We describe a family of l p -norm optimization problems that can be made arbitrarily close to a geometric optimization problem, and show that the dual problems for these approximations are also approximating the dual geometric optimization problem. Finally, we use these approximations and the duality theory for l p -norm optimization to derive simple proofs of the weak and strong duality theorems for geometric optimization.

The author has recently proposed a new way of formulating two classical classes of structured convex problems, geometric and l p -norm optimization, using dedicated convex cones. This approach has some advantages over the traditional formulation: it simpli es the proofs of the well-known associated duality properties (i.e. weak and strong duality) and the design of a polynomial algorithm becomes straightforward.