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Let $\rm{Box}_n = \{x \in \mathbb{R}^n : 0 \leq x \leq e \}$, and let $\rm{QPB}_n$ denote the convex hull of $\{(1, x')'(1, x') : x \in \rm{Box}_n\}$. The quadratic programming problem $\min\{x'Q x + q'x : x \in \rm{Box}_n\}$ where $Q$ is not positive semidefinite (PSD), is equivalent to a linear optimization problem over $\rm{QPB}_n$ and could be efficiently solved if a tractable characterization of $\rm{QPB}_n$ was available. It is known that $\rm{QPB}_2$ can be represented using a PSD constraint combined with constraints generated using the reformulation-linearization technique (RLT). The triangle (TRI) inequalities are also valid for $\rm{QPB}_3$, but the PSD, RLT and TRI constraints together do not fully characterize $\rm{QPB}_3$. In this paper we describe new valid linear inequalities for $\rm{QPB}_n$, $n \geq 3$ based on strengthening the approximation of $\rm{QPB}_3$ given by the PSD, RLT and TRI constraints. These new inequalities are generated in a systematic way using a known disjunctive characterization for $\rm{QPB}_3$. We also describe a conic strengthening of the linear inequalities that incorporates second-order cone constraints. We show computationally that the new inequalities and their conic strengthenings obtain exact solutions for some nonconvex box-constrained instances that are not solved exactly using the PSD, RLT and TRI constraints.

The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by Pippenger and Golumbic. In $2017$, Czabarka, Sz\'ekely and Wagner extended this notion to leaf-labeled rooted binary trees, which are objects widely studied in the field of phylogenetics. They obtain the first results and bounds for the densities and inducibilities of such trees. Following up on their work, we apply Razborov's flag algebra theory to this setting, introducing the flag algebra of rooted leaf-labeled binary trees. This framework allows us to use polynomial optimization methods, based on semidefinite programming, to efficiently obtain new upper bounds for the inducibility of trees and to improve existing ones. Additionally, we obtain the first outer approximations of profiles of trees, which represent all possible simultaneous densities of a pair of trees. Finally, we are able to prove the non-convexity of some of these profiles., Comment: Added code

Given a linear ordering of the vertices of a graph, the cutwidth of a vertex $v$ with respect to this ordering is the number of edges from any vertex before $v$ (including $v$) to any vertex after $v$ in this ordering. The cutwidth of an ordering is the maximum cutwidth of any vertex with respect to this ordering. We are interested in finding the cutwidth of a graph, that is, the minimum cutwidth over all orderings, which is an NP-hard problem. In order to approximate the cutwidth of a given graph, we present a semidefinite relaxation. We identify several classes of valid inequalities and equalities that we use to strengthen the semidefinite relaxation. These classes are on the one hand the well-known 3-dicycle equations and the triangle inequalities and on the other hand we obtain inequalities from the squared linear ordering polytope and via lifting the linear ordering polytope. The solution of the semidefinite program serves to obtain a lower bound and also to construct a feasible solution and thereby having an upper bound on the cutwidth. In order to evaluate the quality of our bounds, we perform numerical experiments on graphs of different sizes and densities. It turns out that we produce high quality bounds for graphs of medium size independent of their density in reasonable time. Compared to that, obtaining bounds for dense instances of the same quality is out of reach for solvers using integer linear programming techniques., Comment: This version differs from the version published in Computers & Operations Research by moving one sentence from Section 2.1 to Section 2.2.1