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 ...

This paper provides a self-contained introduction to the theory of self-concordant functions [8] and applies it to several classes of structured convex optimization problems. We describe the classical short-step interior-point method and optimize its parameters to provide its best possible iteration bound. We also discuss the necessity of introducing two parameters in the definition of self-concordancy, how they react to addition and scaling and which one is the best to fix. A lemma from [2] is improved and allows us to review several classes of structured convex optimization problems and evaluate their algorithmic complexity, using the self-concordancy of the associated logarithmic barriers.

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.

Abstract. The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J.B. Lasserre. Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19, 1995–2014, 2009.] We give a new proof of the finite convergence property, that does not require the assumption that the Hessian of the objective be positive definite on the entire feasible set, but only at the optimal solution. In addition, we show that the number of steps needed for convergence depends on more than the input size of the problem. In particular, the size of the semidefinite program that gives the exact reformulation of the convex polynomial optimization problem may be exponential in the input size. Key words. convex polynomial optimization, sum of squares of polynomials, positivstellensatz, semidefinite programming. AMS subject classifications. 90C60, 90C56, 90C26 1. Polynomial optimization and the Lasserre hierarchy. We consider the polynomial optimization problem pmin =min