Conic optimization: an elegant framework for convex optimization

Abstract

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.
After a brief introduction to convex optimization, the notion of convex cone is introduced, which leads to the conic formulation of convex optimization problems. This formulation features a very symmetric dual problem, and several useful duality theorems pertaining to this conic primal-dual pair are presented. The usefulness of this approach is then demonstrated with its application to a well known class
of convex problems called l_p-norm optimization. A suitably defined convex
cone leads to a conic formulation for this problem, which allows us to derive its dual and the associated weak and strong duality properties in a seamless manner.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. After a brief introduction to convex optimization, the notion of convex cone is introduced, which leads to the conic formulation of convex optimization problems. This formulation features a very symmetric dual problem, and several useful duality theorems pertaining to this conic primal-dual pair are presented.
The usefulness of this approach is then demonstrated with its application to a well known class of convex problems called [p-norm optimization. A suitably defined convex cone leads to a conic formulation for this problem, which allows us to derive its dual and the associated weak and strong duality properties in a seamless manner.