We derive the matrix that minimizes a quadratic criterion subject to linear equality constraints. Since we do not assume that the matrix of the quadratic form criterion is nonsingular, our optimization result is more general than those previously published. We make use of the extended result of this paper to (i) obtain the best linear unbiased estimate of the parameters in a linear regression w...

We survey some unusual eigenvalue problems arising in different applications. We show that all these problems can be cast as problems of estimating quadratic forms. Numerical algorithms based on the well-known Gauss-type quadrature rules and Lanczos process are reviewed for computing these quadratic forms. These algorithms reference the matrix in question only through a matrix-vector product op...

It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical te...

A new approach to minimax MPC for systems with bounded external system disturbances and measurement errors is introduced. It is shown that joint deterministic state estimation and minimax MPC can be written as an optimization problem with linear and quadratic matrix inequalities. By linearizing the quadratic matrix inequality, a semidefinite program is obtained. A simulation study indicates tha...

Operative methods of testing the covariance matrix of the innovation sequence of the Kalman filter are proposed. The quadratic form of the random Wishart matrix is used in this process as monitoring statistic, and the testing problem is reduced to the classical problem of minimization of a quadratic form on the unit sphere. As a result, two algorithms for testing the covariance matrix of the in...

We derive LMI-characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co-positivity. These matrix cones are in fact cones of non-convex quadratic functions that are nonnegative on a certain domain. As a domain, we consider for instance the intersection of a (upper) level-set of a quadratic function and a halfplane. Conseq...

We consider a set of continuous algebraic Riccati equations with indefinite quadratic parts that arise in H∞ control problems. It is well known that the approach for solving such type of equations is proposed in the literature. Two matrix sequences are constructed. Three effective methods are described for computing the matrices of the second sequence, where each matrix is the stabilizing solut...

We present a Krylov subspace–type projection method for a quadratic matrix polynomial λ2I − λA − B that works directly with A and B without going through any linearization. We discuss a special case when one matrix is a low rank perturbation of the other matrix. We also apply the method to solve quadratically constrained linear least squares problem through a reformulation of Gander, Golub, and...

In this paper, the quadratic stabilization of switched linear systems with polytopic uncertainties is considered. Compared with the existing result, a more general switching control method is proposed to guarantee the quadratic stabilization. This switching control method is based on the partition of vertex matrix set of each subsystem. By this method, the matrix inequalities needed to be solve...

In this paper we consider an aggregation technique introduced by Yıldıran [45] to study the convex hull of regions defined by two quadratic inequalities or by a conic quadratic and a quadratic inequality. Yıldıran [45] shows how to characterize the convex hull of open sets defined by two strict quadratic inequalities using Linear Matrix Inequalities (LMI). We show how this aggregation technique...