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In this paper, we study polynomial norms, i.e., norms that are the $d$th root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from $d$th roots of polynomials, we prove that any norm can be approximated arbitrarily well by a polynomial norm. We then investigate the computational problem of testing whether a form gives a polynomial norm. We show that this problem is strongly NP-hard already when the degree of the form is 4, but can always be answered by solving a hierarchy of semidefinite programs. We further study the problem of optimizing over the set of polynomial norms using semidefinite programming. To do this, we introduce the notion of $r$-sum of squares-convexity and extend a result of Reznick on sum of squares representations of positive definite forms to positive definite biforms. We conclude with some applications of polynomial norms to statistics and dynamical systems.

The goal of this thesis is to show how to derive in a completely automated way exact and global worst-case guarantees for first-order methods in convex optimization. To this end, we formulate a generic optimization problem looking for the worst-case scenarios. The worst-case computation problems, referred to as performance estimation problems (PEPs), are intrinsically infinite-dimensional optimization problems formulated over a given class of objective functions. To render those problems tractable, we develop (smooth and non-smooth) convex interpolation framework, which provides necessary and sufficient conditions to interpolate our objective functions. With this idea, we transform PEPs into solvable finite-dimensional semidefinite programs, from which one obtains worst-case guarantees and worst-case functions, along with the corresponding explicit proofs. PEPs already proved themselves very useful as a tool for developing convergence analyses of first-order optimization methods. Among others, PEPs allow obtaining exact guarantees for gradient methods, along with their inexact, projected, proximal, conditional, decentralized and accelerated versions., (FSA - Sciences de l'ingénieur) -- UCL, 2017

We consider the problem of minimizing a given $n$-variate polynomial $f$ over the hypercube $[-1,1]^n$. An idea introduced by Lasserre, is to find a probability distribution on $[-1,1]^n$ with polynomial density function $h$ (of given degree $r$) that minimizes the expectation $\int_{[-1,1]^n} f(x)h(x)d\mu(x)$, where $d\mu(x)$ is a fixed, finite Borel measure supported on $[-1,1]^n$. It is known that, for the Lebesgue measure $d\mu(x) = dx$, one may show an error bound $O(1/\sqrt{r})$ if $h$ is a sum-of-squares density, and an $O(1/r)$ error bound if $h$ is the density of a beta distribution. In this paper, we show an error bound of $O(1/r^2)$, if $d\mu(x) = \left( \prod_{i=1}^n \sqrt{1-x_i^2} \right)^{-1}$ (the well-known measure in the study of orthogonal polynomials), and $h$ has a Schmüdgen-type representation with respect to $[-1,1]^n$, which is a more general condition than a sum of squares. The convergence rate analysis relies on the theory of polynomial kernels and, in particular, on Jackson kernels. We also show that the resulting upper bounds may be computed as generalized eigenvalue problems, as is also the case for sum-of-squares densities.

We consider the gradient (or steepest) descent method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadratic function. We also extend the result to a noisy variant of gradient descent method, where exact line-search is performed in a search direction that differs from negative gradient by at most a prescribed relative tolerance. The proof is computer-assisted, and relies on the resolution of semidefinite programming performance estimation problems as introduced in the paper [Y. Drori and M. Teboulle. Performance of first-order methods for smooth convex minimization: a novel approach. Mathematical Programming, 145(1-2):451-482, 2014].

We consider the gradient (or steepest) descent method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadratic function. We also give the tight worst-case complexity bound for a noisy variant of gradient descent method, where exact line-search is performed in a search direction that differs from negative gradient by at most a prescribed relative tolerance. The proofs are computer-assisted, and rely on the resolutions of semidefinite programming performance estimation problems as introduced in the paper [Y. Drori and M. Teboulle. Performance of first-order methods for smooth convex minimization: a novel approach. Mathematical Programming, 145(1-2):451-482, 2014].

It is known that one can solve semidefinite programs to within fixed accuracy in polynomial time using the ellipsoid method (under some assumptions). In this paper it is shown that the same holds true when one uses the short step, primal interior point method. The main idea of the proof is to employ Diophantine approximation at each iteration to bound the intermediate bit sizes of iterates.