Related papers: The complete edge relaxation for binary polynomial…
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic…
Edge ideals of finite simple graphs are well-studied over polynomial rings. In this paper, we initiate the study of edge ideals over exterior algebras, specifically focusing on the depth and singular varieties of such ideals. We prove an…
In this paper we introduce a natural model for the realization space of a polytope up to projective equivalence which we call the slack realization space of the polytope. The model arises from the positive part of an algebraic variety…
In this paper, we study the strength of convex relaxations obtained by convexification of aggregation of constraints for a set $S$ described by two bilinear bipartite equalities. Aggregation is the process of rescaling the original…
In this paper we address the graph matching problem. Following the recent works of \cite{zaslavskiy2009path,Vestner2017} we analyze and generalize the idea of concave relaxations. We introduce the concepts of conditionally concave and…
The subject of this contribution is the study of the Lov\'asz-Schrijver PSD-operator N+ applied to the edge relaxation of the stable set polytope of a graph. We are particularly interested in the problem of characterizing graphs for which…
We consider the general polynomial optimization problem $P: f^*=\min \{f(x)\,:\,x\in K\}$ where $K$ is a compact basic semi-algebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the…
We introduce a semidefinite relaxation for optimal control of linear systems with time scaling. These problems are inherently nonconvex, since the system dynamics involves bilinear products between the discretization time step and the…
We develop the tools to bound extreme roots of multivariate real zero polynomials globally. This is done through the use of a relaxation that approximates their rigidly convex sets. This relaxation can easily be constructed using the degree…
We obtain an improved lower bound for the regularity of the binomial edge ideals of trees. We prove an upper bound for the regularity of the binomial edge ideals of certain subclass of block-graphs. As a consequence we obtain sharp upper…
Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. We consider this property for regular polytopes in arbitrary dimensions, notably the simplex, cube, and…
In this paper we extend the dichotomy given by Samuelsson and Wold that can be thought of as an analogue of the Wermer maximality theorem in $\mathbb{C}^2$ for certain polynomial polyhedra. We consider complex non-degenerate simply…
In Speakman and Lee (2017), we analytically developed the idea of using volume as a measure for comparing relaxations in the context of spatial branch-and-bound. Specifically, for trilinear monomials, we analytically compared the three…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
Motivated by the analysis of the performance of the simplex method we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules which are fundamental for the following reasons: First, they…
Globally optimizing a nonconvex quadratic over the intersection of $m$ balls in $\mathbb{R}^n$ is known to be polynomial-time solvable for fixed $m$. Moreover, when $m=1$, the standard semidefinite relaxation is exact. When $m=2$, it has…
We study the maximization of sums of heterogeneous quadratic forms over the Stiefel manifold, a nonconvex problem that arises in several modern signal processing and machine learning applications such as heteroscedastic probabilistic…
We investigate the Sherali-Adams lift & project hierarchy applied to a graph isomorphism polytope whose integer points encode the isomorphisms between two graphs. In particular, the Sherali-Adams relaxations characterize a new vertex…
It is known that every lattice polytope is unimodularly equivalent to a face of some reflexive polytope. A stronger question is to ask whether every $(0,1)$-polytope is unimodularly equivalent to a facet of some reflexive polytope. A large…
In this paper, we study a broad class of fully nonlinear elliptic equations on Hermitian manifolds. On one hand, under the optimal structural assumptions we derive $C^{2,\alpha}$-estimate for solutions of the equations on closed Hermitian…