In this paper, we formulate the l p -norm optimization problem as a conic optimization problem, derive its standard duality properties and show it can be solved in polynomial time. We first define an ad hoc closed convex cone L p , study its properties and derive its dual. This allows us to express the standard l p -norm optimization primal problem as a conic problem involving L p . Using convex conic duality and our knowledge about L p , we proceed to derive the dual of this problem and prove the well-known regularity properties of this primal-dual pair, i.e. zero duality gap and primal attainment. Finally, we prove that the class of l p -norm optimization problems can be solved up to a given accuracy in polynomial time, using the framework of interior-point algorithms and self-concordant barriers.

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 purpose of this survey article is to introduce the reader to a very elegant formulation of convex optimization problems called conic optimization and outline its many advantages.