Trust Your Data or Not—StQP Remains StQP: Community Detection via Robust Standard Quadratic Optimization

We consider the robust standard quadratic optimization problem (RStQP), in which an uncertain (possibly indefinite) quadratic form is optimized over the standard simplex. Following most approaches, we model the uncertainty sets by balls, polyhedra, or spectrahedra, more generally, by ellipsoids or order intervals intersected with subcones of the copositive matrix cone. We show that the copositive relaxation gap of the RStQP equals the minimax gap under some mild assumptions on the curvature of the aforementioned uncertainty sets and present conditions under which the RStQP reduces to the standard quadratic optimization problem. These conditions also ensure that the copositive relaxation of an RStQP is exact. The theoretical findings are accompanied by the results of computational experiments for a specific application from the domain of graph clustering, more precisely, community detection in (social) networks. The results indicate that the cardinality of communities tend to increase for ellipsoidal uncertainty sets and to decrease for spectrahedral uncertainty sets.

On Restriction Estimates for Discrete Quadratic Surfaces

We obtain truncated restriction estimates of an unexpected form for discrete surfaces $$\begin{align*}S_N = \{\, ( n_1 , \dots , n_d , R( n_1 , \dots, n_d ) ) \,,\, n_i \in [-N,N] \cap {\mathbb{Z}} \,\},\end{align*}$$ where $R$ is an indefinite quadratic form with integer matrix.

H∞Fault Detection for Linear Discrete Time-Varying Descriptor Systems with Missing Measurements

This paper deals with the problem ofH∞fault detection for a class of linear discrete time-varying descriptor systems with missing measurements, and the missing measurements are described by a Bernoulli random binary switching sequence. We first translate theH∞fault detection problem into an indefinite quadratic form problem. Then, a sufficient and necessary condition on the existence of the minimum is derived. Finally, an observer-basedH∞fault detection filter is obtained such that the minimum is positive and its parameter matrices are calculated recursively by solving a matrix differential equation. A numerical example is given to demonstrate the efficiency of the proposed method.

H∞Estimation for a Class of Lipschitz Nonlinear Discrete-Time Systems with Time Delay

The issue ofH∞estimation for a class of Lipschitz nonlinear discrete-time systems with time delay and disturbance input is addressed. First, through integrating theH∞filtering performance index with the Lipschitz conditions of the nonlinearity, the design of robust estimator is formulated as a positive minimum problem of indefinite quadratic form. Then, by introducing the Krein space model and applying innovation analysis approach, the minimum of the indefinite quadratic form is obtained in terms of innovation sequence. Finally, through guaranteeing the positivity of the minimum, a sufficient condition for the existence of theH∞estimator is proposed and the estimator is derived in terms of Riccati-like difference equations. The proposed algorithm is proved to be effective by a numerical example.

INSTABILITY IN THE PRESENCE OF AN INDEFINITE CONSERVATION LAW

Consider a C1 vector field X on a finite-dimensional manifold M and xe an equilibrium point for the dynamics [Formula: see text]. We will prove that if I is a C2 constant of motion for X such that xe is also a critical point of I, then [Formula: see text] is a constant of motion for the linearized system [Formula: see text]. As a consequence we will give an instability result under the condition that [Formula: see text] is an indefinite quadratic form. If xe is not a critical point of I, then we obtain [Formula: see text] as a constant of motion for the linearized system.

Localized Elasticae for the Strut on the Linear Foundation

Localized solutions, for the classical problem of the nonlinear strut (elastica) on the linear elastic foundation, are predicted from double-scale analysis, and confirmed from nonlinear volume-preserving Runge-Kutta runs. The dynamical phase-space analogy introduces a spatial Lagrangian function, valid over the initial post-buckling range, with kinetic and potential energy components. The indefinite quadratic form of the spatial kinetic energy admits unbounded solutions, corresponding to escape from a potential well. Numerical experimentation demonstrates that there is a fractal edge to the escape boundary, resulting in spatial chaos.

Comparative lagrangian formulations for localized buckling

An energy functional for a strut on a nonlinear softening foundation is worked into two different lagrangian forms, in fast and slow space respectively. The developments originate independently of the underlying differential equation, and carry some quite general features. In each case, the kinetic energy is an indefinite quadratic form. In fast space, this leads to an escape phenomenon with fractal properties. In slow space, kinetic energy is added to a potential contribution that is familiar from modal formulations. Together, and in conjunction with a recent set of numerical experiments, they illustrate the extra complexities of localized, as opposed to distributed periodic, buckling.