Related papers: Density, Determinacy, Duality and a Regularized Mo…
Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficiencies related to global constraints. First, they were not able to support a "cardinality constraint", as in, say, the Min-Bisection problem. Second, while the…
This paper presents algorithms that upper-bound the peak value of a state function along trajectories of a continuous-time system with rational dynamics. The finite-dimensional but nonconvex peak estimation problem is cast as a convex…
This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to…
A variant of the Archimedean Positivstellensatz is proved which is based on Archimedean semirings or quadratic modules of generating subalgebras. It allows one to obtain representations of strictly positive polynomials on compact…
We use moment-SOS (Sum Of Squares) relaxations to address the optimal control problem of the 1D heat equation perturbed with a nonlinear term. We extend the current framework of moment-based optimal control of PDEs to consider a quadratic…
We prove that, under some additional assumption, Putinar's Positivstellensatz holds on cylinders of type $S \times {\mathbb R}$ with $S = \{x \in {\mathbb R}^n | g_1(x) \ge 0, ..., g_s(x) \ge 0\}$ such that the quadratic module generated by…
Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can…
We introduce a convergent hierarchy of lower bounds on the minimum value of a real form over the unit sphere. The main practical advantage of our hierarchy over the real sum-of-squares (RSOS) hierarchy is that the lower bound at each level…
We introduce Stochastic Asymptotical Regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed linear-operator equations, which are deterministic models for numerous inverse problems in…
Let $\tau$ be a locally convex topology on the countable dimensional polynomial $\reals$-algebra $\rx:=\reals[X_1,...,X_n]$. Let $K$ be a closed subset of $\reals^n$, and let $M:=M_{\{g_1, ... g_s\}}$ be a finitely generated quadratic…
The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure for constructing a sequence of increasingly tight semidefinite relaxations. It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the…
The arithmetic regularity lemma for $\mathbb{F}_p^n$, proved by Green in 2005, states that given a subset $A\subseteq \mathbb{F}_p^n$, there exists a subspace $H\leq \mathbb{F}_p^n$ of bounded codimension such that $A$ is Fourier-uniform…
The Sum of Squares (\sos{}) hierarchy gives an automatized technique to create a family of increasingly tight convex relaxations for binary programs. There are several problems for which a constant number of rounds of this hierarchy give…
In this paper, we study regular sets in metric measure spaces with bounded Ricci curvature. We prove that the existence of a point in the regular set of the highest dimension implies the positivity of the measure of such regular set. Also…
Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size k is planted in a random G(n, 1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best…
Based on a compactness method, we establish regularity criteria for suitable weak solutions to the surface growth model with a forcing term. These criteria imply that the H\"older regularity of solutions follows from smallness conditions on…
We present a hierarchy of semidefinite programs (SDPs) for the problem of fitting a shape-constrained (multivariate) polynomial to noisy evaluations of an unknown shape-constrained function. These shape constraints include convexity or…
Finding a global solution to the optimal power flow (OPF) problem is difficult due to its nonconvexity. A convex relaxation in the form of semidefinite programming (SDP) has attracted much attention lately as it yields a global solution in…
We prove a general quantitative theorem on the asymptotic behavior of stochastic quasi-Fej\'er monotone sequences in a broad metric context. Concretely, our result explicitly constructs a rate of convergence for such process, both in mean…
Shape constraints, such as non-negativity, monotonicity, convexity or supermodularity, play a key role in various applications of machine learning and statistics. However, incorporating this side information into predictive models in a hard…