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.

In this paper, we formulate the l p -norm optimization problem as a conic optimization problem and derive its standard duality properties. We first define an ad hoc closed convex cone L and derive its dual. We express then the standard l p -norm optimization primal problem as a conic problem involving L. Using convex conic duality, we derive the dual of this problem and prove the well-known regularity properties of this primal-dual pair, i.e. zero duality gap and dual attainment.

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 paper is to provide improved complexity results for several classes of structured convex optimization problems using to the theory of self-concordant functions developed in [11]. We describe the classical short-step interior-point method and optimize its parameters in order to provide the best possible iteration bound. We also discuss the necessity of introducing two parameters in the definition of self-concordancy and which one is the best to fix. A lemma from [3] is improved, which allows us to review several classes of structured convex optimization problems and improve the corresponding complexity results.

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.

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