Solvers for the verified solution of parametric linear systems

2015

J Glob Optim

The classical αBB method is a global optimization method the important step of which is to determine a convex underestimator of an objective function on an interval domain. Its particular point is to enclose the range of a Hessian matrix in an interval matrix. To have a tighter estimation of the Hessian matrices, we investigate a linear parametric form enclosure in this paper. One way to obtain this form is by using a slope extension of the Hessian entries. Numerical examples indicate that our approach can sometimes significantly reduce overestimation on the objective function. However, the slope extensions highly depend on a choice of the center of linearization. We compare some naive choices and also propose a heuristic one, which performs well in executed examples, but it seems there is no one global winner.

Section: Hessians In Linear Parametric Formsmentioning

confidence: 98%

An extension of the $$\alpha \hbox {BB}$$ α BB -type underestimation to linear parametric Hessian matrices

2015

J Glob Optim

“…Consider interval parametric linear system (27), which occurs in worst-case tolerance analysis of linear AC (alternate current) electrical circuits. [48][49][50][51] The circuit studied is shown in Figure 6 (cf. the work of Kolev 48 ).…”

Section: Example 14mentioning

confidence: 99%

Direct and iterative methods for interval parametric algebraic systems producing parametric solutions

Skalna

Numerical Linear Algebra App

2019

Summary This paper deals with interval parametric linear systems with general dependencies. Motivated by the so‐called parameterized solution introduced by Kolev, we consider the enclosures of the solution set in a revised affine form. This form is advantageous to a classical interval solution because it enables us to obtain both outer and inner bounds for the parametric solution set and, thus, intervals containing the endpoints of the hull solution, among others. We propose two solution methods, a direct method called the generalized expansion method and an iterative method based on interval‐affine Krawczyk iterations. For the iterative method, we discuss its convergence and show the respective sufficient criterion. For both methods, we perform theoretical and numerical comparisons with some other approaches. The numerical experiments, including also interval parametric linear systems arising in practical problems of structural and electrical engineering, indicate the great usefulness of the proposed methodology and its superiority over most of the existing approaches to solving interval parametric linear systems.

“…Linear parametric forms are also used to model linear dependencies between parameters in interval linear equation solving [10,30,31,41]. Linear dependencies cause not only the problem to be more difficult from the computational viewpoint, but it is also hard to describe the corresponding solution set; see Mayer [23].…”

Section: Example 1 Letmentioning

confidence: 99%

Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

Studies in Systems, Decision and Control

2017

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equivalent conditions, and also propose some computationally cheap sufficient / necessary conditions. Our results extend the classical results on positive (semi-)definiteness of interval matrices. They may be useful for checking convexity or non-convexity in global optimization methods based on branch and bound framework and using interval techniques.