Related papers: Polynomial Optimization, Certificates of Positivit…
This paper introduces a new storage-optimal first-order method (FOM), CertSDP, for solving a special class of semidefinite programs (SDPs) to high accuracy. The class of SDPs that we consider, the exact QMP-like SDPs, is characterized by…
We describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of Borel measurable functions, with particular emphasis to algebras generated by semi-algebraic functions. In which case the standard…
Certificates of polynomial nonnegativity can be used to obtain tight dual bounds for polynomial optimization problems. We consider Sums of Nonnegative Circuit (SONC) polynomials certificates, which are well suited for sparse problems since…
This paper investigates the minimization of the expectation of piecewise polynomial loss functions over Wasserstein balls. This optimization problem often appears as a key sub-problem of distributionally robust optimization problems. We…
This paper studies the matrix Moment-SOS hierarchy for solving polynomial matrix optimization. Our first result is to show the finite convergence of this hierarchy, if the nondegeneracy condition, strict complementarity condition and second…
A reduced-bias nonparametric estimator of the cumulative distribution function (CDF) and the survival function is proposed using infinite-order kernels. Fourier transform theory on generalized functions is utilized to obtain the improved…
We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semi-algebraic) design spaces. We use the moment-sum-of-squares hierarchy of semidefinite programming problems…
This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the…
This paper is concerned with polynomial optimization problems. We show how to exploit term (or monomial) sparsity of the input polynomials to obtain a new converging hierarchy of semidefinite programming relaxations. The novelty (and…
We present a new approach to the design of D-optimal experiments with multivariate polynomial regressions on compact semi-algebraic design spaces. We apply the moment-sum-of-squares hierarchy of semidefinite programming problems to solve…
We introduce an S.o.S hierarchy of lower bounds for a polynomial optimization problem whose constraint is expressed as a matrix polynomial semidefinite inequality. Our approach involves utilizing a penalty function framework to directly…
We introduce a new framework for unifying and systematizing the performance analysis of first-order black-box optimization algorithms for unconstrained convex minimization. The low-cost iteration complexity enjoyed by first-order algorithms…
We report the experimental results on certifying 1% global optimality of solutions of AC-OPF instances from PGLiB via the CS-TSSOS hierarchy -- a moment-SOS based hierarchy that exploits both correlative and term sparsity, which can provide…
We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial…
Certificates of non-negativity such as Putinar's Positivstellensatz have been used to obtain powerful numerical techniques to solve polynomial optimization (PO) problems. Putinar's certificate uses sum-of-squares (sos) polynomials to…
This paper establishes new Positivstellens\"atze for polynomials that are positive on sets defined by polynomial matrix inequalities (PMIs). We extend the classical Handelman and Krivine-Stengle theorems from the scalar inequality setting…
We consider a hierarchy of upper approximations for the minimization of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$ proposed recently by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the push-forward…
This chapter investigates how symmetries can be used to reduce the computational complexity in polynomial optimization problems. A focus will be specifically given on the Moment-SOS hierarchy in polynomial optimization, where results from…
We construct near-optimal coresets for kernel density estimates for points in $\mathbb{R}^d$ when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size $O(\sqrt{d}/\varepsilon\cdot…
We study the problem of {\sl certification}: given queries to a function $f : \{0,1\}^n \to \{0,1\}$ with certificate complexity $\le k$ and an input $x^\star$, output a size-$k$ certificate for $f$'s value on $x^\star$. This abstractly…