Related papers: Extreme Nonnegative Quadratics over Stanley Reisne…
One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the formulas naturally contain…
This paper presents two novel algorithms for approximately projecting symmetric matrices onto the Positive Semidefinite (PSD) cone using Randomized Numerical Linear Algebra (RNLA). Classical PSD projection methods rely on full-rank…
We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD) slack. Beyond that, we show that if the…
We study the problem of estimating a low-rank positive semidefinite (PSD) matrix from a set of rank-one measurements using sensing vectors composed of i.i.d. standard Gaussian entries, which are possibly corrupted by arbitrary outliers.…
We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are…
Given underdetermined measurements of a Positive Semi-Definite (PSD) matrix $X$ of known low rank $K$, we present a new algorithm to estimate $X$ based on recent advances in non-convex optimization schemes. We apply this in particular to…
This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex…
Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or…
This paper investgates Stanley-Reisner ideals with pure resolutions. We first describe two infinite families of such ideals associated to highly symmetric complexes. We then prove a partial analogue to the first Boij-S\"oderberg Conjecture…
We study the problem of testing whether a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ with bounded entries ($\|\mathbf{A}\|_\infty \leq 1$) is positive semi-definite (PSD), or $\epsilon$-far in Euclidean distance from the PSD cone,…
This paper presents various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at…
A finite projective plane, or more generally a finite linear space, has an associated incidence complex that gives rise to two natural algebras: the Stanley-Reisner ring $R/I_\Lambda$ and the inverse system algebra $R/I_\Delta$. We give a…
We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2. We notably show, that such a variety $X…
We characterize the variety of compatible fundamental matrix triples by computing its multidegree and multihomogeneous vanishing ideal. This answers the first interesting case of a question recently posed by Br{\aa}telund and Rydell. Our…
Let $X \subset \mathbb{P}^{n+c}$ be a nondegenerate smooth projective variety of dimension $n$ defined by quadratic equations. For such varieties, P. Ionescu and F. Russo proved the Hartshorne conjecture on complete intersections, which…
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others.…
We give a complete characterization of the holonomies of strictly convex cusps and of round cusps in convex projective geometry. We build families of generalized cusps of non-maximal rank associated to each strictly convex or round cusp. We…
A weakly infeasible semidefinite program (SDP) has no feasible solution, but it has approximate solutions whose constraint violation is arbitrarily small. These SDPs are ill-posed and numerically often unsolvable. They are also closely…
We investigate the properties of a class of piecewise-fractional maps arising from the introduction of an invariance under rescaling into convex quadratic maps. The subsequent maps are quasiconvex, and pseudoconvex on specific convex cones;…
We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of for00504925mulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in…