9 research outputs found
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
VyuĹľitĂ branch and bound pĹ™Ăstupu pro parametrickĂ© intervalovĂ© lineárnĂ 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 zaobĂrá parametrickĂ˝mi intervalovĂ˝mi lineárnĂmi soustavami. Branch and bound metodou a rĹŻznĂ˝mi námi implementovanĂ˝mi proĹ™ezávacĂmi podmĂnkami jsme dostali jejich mnoĹľinu Ĺ™ešenĂ. PĹ™esnÄ›ji jsme ji popsali po- mocĂ n-rozmÄ›rnĂ˝ch boxĹŻ, kterĂ© jsme zĂskali dĂky vyuĹľitĂ˝m metodám. Seznámili jsme se se základnĂmi pojmy ohlednÄ› intervalĹŻ a lineárnĂch soustav. NáslednÄ› jsme zpracovávali zĂskanĂ© boxy Ĺ™ešenĂ vĂcerĂ˝mi metodami s cĂlem optimali- zovat jejich poÄŤet. SoučástĂ práce je i porovnánĂ jednotlivĂ˝ch proĹ™ezávacĂch podmĂnek na parametrickĂ˝ch soustavách s rĹŻznĂ˝m poÄŤtem parametrĹŻ. Uve- denĂ© algoritmy byly implementovanĂ© do intervalovĂ©ho balĂku Lime s moĹľnostĂ jednoduchĂ© vizualizace zĂskanĂ˝ch Ĺ™ešenĂ. 1Katedra aplikovanĂ© matematikyDepartment of Applied MathematicsFaculty of Mathematics and PhysicsMatematicko-fyzikálnĂ fakult