相关论文: Polynomial Optimization, Certificates of Positivit…
We consider the global minimization of a polynomial on a compact set B. We show that each step of the Moment-SOS hierarchy has a nice and simple interpretation that complements the usual one. Namely, it computes coefficients of a polynomial…
Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set ${\mathbf{X} \subseteq \mathbb{R}^n}$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on…
The classical Moment-Sum Of Squares hierarchy allows to approximate a global minimum of a polynomial optimization problem through semidefinite relaxations of increasing size. However, for many optimization instances, solving higher order…
We study sum-of-squares (SOS) certificates for nonnegative polynomials $p$ on $\mathbb{R}^d$ and their implications for polynomial optimization over unbounded domains. Building on Lasserre's perturbation approach, we consider SOS…
We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of "the CF…
The Schm\"udgen's Positivstellensatz gives a certificate to verify positivity of a strictly positive polynomial $f$ on a compact, basic, semi-algebraic set $\mathbf{K} \subset \mathbb{R}^n$. A Positivstellensatz of this type is called…
We introduce mollified Christoffel-Darboux (CD) kernels on varieties, a systematic regularization of the classical CD kernel associated with a probability measure on a compact domain. The main motivations are twofold: first, to sharpen the…
In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially…
We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem $(P):\:f^{\ast}=\min \{\,f(x):x\in K\,\}$ on a compact basic semi-algebraic set $K\subset\R^n$. This hierarchy combines some advantages…
One considers polynomial optimization problems with compact feasible set $\mathbf{\Omega}$ defined by SOS-concave polynomials $g_j$, and with a globally non-convex polynomial objective $f$. We show that if $f$ is strongly convex on…
We consider certificates of positivity for univariate polynomials with rational coefficients that are positive over (an interval of)~$\mathbb{R}$. Such certificates take the form of weighted sums of squares (SOS) of polynomials with…
A polynomial that is a sum of squares (SOS) of other polynomials is evidently positive. The converse is not true, there are positive polynomials which are not SOS. This note focuses on the problem of certifying, in exact arithmetic, that a…
The standard moment-sum-of-squares (SOS) hierarchy is a powerful method for solving global polynomial optimization problems. However, its convergence relies on Putinar's Positivstellensatz, which requires the feasible set to satisfy the…
We revisit Stengle's classical univariate polynomial optimization example $min 1 - x^2 s.t. (1 - x^2)^3 \geq 0$ whose constraint description is degenerate at the minimizers. We prove that the moment-SOS hierarchy of relaxation order $r \geq…
We investigate a class of moment problems, namely recovering a measure supported on the graph of a function from partial knowledge of its moments, as for instance in some problems of optimal transport or density estimation. We show that the…
The Moment-SOS hierarchy initially introduced in optimization in 2000, is based on the theory of the K-moment problem and its dual counterpart, polynomials that are positive on K. It turns out that this methodology can be also applied to…
Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems are based on sums…
Let $p_{\min}$ denote the minimum of a polynomial $p$ over a (general) compact semialgebraic set $S \subseteq \mathbb{R}^n$. A standard way to approximate $p_{\min}$ is via hierarchies built from Positivstellens\"atze, which certify…
We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of…
We study a class of polynomial optimization problems with a robust polynomial matrix inequality (PMI) constraint where the uncertainty set itself is defined also by a PMI. These can be viewed as matrix generalizations of semi-infinite…