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In the past decade or so, semi-definite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of fundamental importance because its ultimate goal of constructing or… (More)

Any given nonnegative matrix A ∈ R m×n can be expressed as the product A = U V for some nonneg-ative matrices U ∈ R m×k and V ∈ R k×n with k ≤ min{m, n}. The smallest k that makes this factorization possible is called the nonnegative rank of A. Computing the exact nonnegative rank and the corresponding factorization are known to be NP-hard. Even if the… (More)

—Matrix factorization has been of fundamental importance in modern sciences and technology. This work investigates the notion of factorization with entries restricted to integers or binaries, , where the " integer " could be either the regular ordinal integers or just some nominal labels. Being discrete in nature, such a factorization or approximation… (More)

In this study, nonnegative matrix factorization is recast as the problem of approximating a polytope on the probability simplex by another polytope with fewer facets. Working on the probability simplex has the advantage that data are limited to a compact set with known boundary, making it easier to trace the approximation procedure. In particular, the… (More)

The Euclidean distance matrix for n distinct points in ℝ (r) is generically of rank r + 2. It is shown in this paper via a geometric argument that its nonnegative rank for the case r = 1 is generically n.

Many natural phenomena can be modeled by a second-order dynamical system M ¨ y +C ˙ y +Ky = f (t), where y(t) stands for an appropriate state variable and M , C, K are time-invariant, real and symmetric matrices. In contrast to the classical inverse vibration problem where a model is to be determined from natural frequencies corresponding to various… (More)

Bounded, semi-infinite Hankel matrices of finite rank over the space 2 of square-summable sequences occur frequently in classical analysis and engineering applications. The notion of finite rank often appears under different contexts and the literature is diverse. The first part of this paper reviews some elegant, classical criteria and establishes… (More)

Modern economic theory views the economy as a dynamical system in which rational decisions are made in the face of uncertainties. The dynamics includes changes over time of market behavior such as consumption, investment, labor supply, and technology innovation, all interpreted in a broad sense. The Euler equation arises as the first order optimality… (More)

Quadratic pencils arise in many areas of important applications. The underlying physical systems often impose inherent structures, which include the predetermined inner-connectivity among elements within the physical system and the mandatory nonnegativity of physical parameters, on the pencils. In the inverse problem of reconstructing a quadratic pencil… (More)