Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

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

Centered solutions for uncertain linear equations

Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we derive convex representations for united and tolerable solution sets. Secondly, to obtain centered solutions for uncertain linear equations, we develop a new method based on adjustable robust optimization (ARO) techniques to compute the maximum size inscribed convex body (MCB) of the set of the solutions. In general, the obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We use recent results from ARO to characterize for which convex bodies the obtained MCB is optimal. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input–output model, Colley’s Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method

Využití branch and bound přístupu pro parametrické intervalové lineární soustavy

This work is focused on parametric interval linear systems. By using branch and bound method and various pruning conditions, we first obtained their solution and then described it more precisely with n-dimensional boxes. We were acquainted with the basic concepts of intervals and linear systems. Subsequently, we processed the boxes obtained by multiple methods to opti- mize their number. Part of the work is also a comparison of various pruning conditions on parametric systems with the different number of parameters. Finally, our algorithms were implemented into the Lime interval package with the possibility of simple visualization of the obtained solutions. 1Tato práce se zaobírá parametrickými intervalovými lineárními soustavami. Branch and bound metodou a různými námi implementovanými prořezávacími podmínkami jsme dostali jejich množinu řešení. Přesněji jsme ji popsali po- mocí n-rozměrných boxů, které jsme získali díky využitým metodám. Seznámili jsme se se základními pojmy ohledně intervalů a lineárních soustav. Následně jsme zpracovávali získané boxy řešení vícerými metodami s cílem optimali- zovat jejich počet. Součástí práce je i porovnání jednotlivých prořezávacích podmínek na parametrických soustavách s různým počtem parametrů. Uve- dené algoritmy byly implementované do intervalového balíku Lime s možností jednoduché vizualizace získaných řešení. 1Katedra aplikované matematikyDepartment of Applied MathematicsFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult