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Overdetermined systems of interval linear equations
Horáček, Jaroslav ; Hladík, Milan (advisor)
This work is focused on overdetermined systems of interval linear equati- ons. First part consists of introduction to interval arithmetics and interval linear algebra and basic theory of interval linear systems. In the second part various methods for solving overdetermined interval linear systems are de- scribed. By solution of overdetermined interval system we mean union of all solutions of all subsystems. Known and our variants of algorithms are discussed. We introduce our subsquare method. All mentioned methods are implemented in one toolbox for Matlab. Methods are tested on solvable and unsolvable overdetermined systems. For solvable systems we test solution enclosure, time and special features of methods. For unsolvable systems we test detection of unsolvability. At the end of this work we provide basic in- troduction to Intlab. 1
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Methods for a computation of the optimal value range in interval linear programming
Král, Ondřej ; Hladík, Milan (advisor) ; Novotná, Jana (referee)
This thesis is about the problem of searching an interval that enclose all op- timal values of the objective function in interval linear programming, so called the optimal value range. The solution to this problem is sometimes reduced to solving just a few linear programs but in general it is a hard problem. Af- ter we get familiar with interval arithmetics and when we extend it to linear programming, we define important sets and their properties, B-stability and other connected subproblems. We will extend B-stability to generalized interval linear programming and we will examine methods for computing the optimal value range and we will compare them numerically on random systems. The goal is to implement all mentioned methods in MATLAB/INTLAB and based on numerical results provide one function that will solve this problem, possible efficiently. 1
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Numerical range of an interval matrix
Ivičič, Michal ; Hladík, Milan (advisor) ; Tichý, Petr (referee)
The numerical range of a matrix is a set of complex numbers that contains all the eigen- values of the matrix. It is used for instance to estimate a matrix norm. This thesis is about the numerical range of an interval matrix. In the theoretical part, we examine its properties. We prove for example that it is NP-hard to find out whether a given point lies in the numerical range. On an example, we show that field of values of an interval matrix is not necessarily convex. The thesis contains descriptions of two algorithms for visualization of the convex hull of the numerical range. Both of them are only suitable for matrices of small sizes due to high time complexity. Therefore we also present a polyno- mial algorithm for computing the upper bound of the numerical range. In the practical part, we implement the algorithms as functions in the Matlab language. 1
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Interval linear and nonlinear systems
Horáček, Jaroslav ; Hladík, Milan (advisor) ; Garloff, Jürgen (referee) ; Ratschan, Stefan (referee)
First, basic aspects of interval analysis, roles of intervals and their applications are addressed. Then, various classes of interval matrices are described and their relations are depicted. This material forms a prelude to the unifying theme of the rest of the work - solving interval linear systems. Several methods for enclosing the solution set of square and overdetermined interval linear systems are covered and compared. For square systems the new shaving method is introduced, for overdetermined systems the new subsquares approach is introduced. Detecting unsolvability and solvability of such systems is discussed and several polynomial conditions are compared. Two strongest condi- tions are proved to be equivalent under certain assumption. Solving of interval linear systems is used to approach other problems in the rest of the work. Computing enclosures of determinants of interval matrices is addressed. NP- hardness of both relative and absolute approximation is proved. New method based on solving square interval linear systems and Cramer's rule is designed. Various classes of matrices with polynomially computable bounds on determinant are characterized. Solving of interval linear systems is also used to compute the least squares linear and nonlinear interval regression. It is then applied to real...
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Monge property for interval matrices
Černý, Martin ; Hladík, Milan (advisor) ; Zimmermann, Karel (referee)
The thesis is a first survey in the field of interval matrices with Monge prop- erty. We investigate characteristics and properties of two classes of matrices - interval strong and interval weak Monge matrices. We develop a recognition and reconstruction algorithms and study applications in combinatorial problems and in a problem connected with computational geometry.
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Overdetermined systems of interval linear equations
Horáček, Jaroslav ; Hladík, Milan (advisor)
This work is focused on overdetermined systems of interval linear equati- ons. First part consists of introduction to interval arithmetics and interval linear algebra and basic theory of interval linear systems. In the second part various methods for solving overdetermined interval linear systems are de- scribed. By solution of overdetermined interval system we mean union of all solutions of all subsystems. Known and our variants of algorithms are discussed. We introduce our subsquare method. All mentioned methods are implemented in one toolbox for Matlab. Methods are tested on solvable and unsolvable overdetermined systems. For solvable systems we test solution enclosure, time and special features of methods. For unsolvable systems we test detection of unsolvability. At the end of this work we provide basic in- troduction to Intlab. 1
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Computational Bounded Rationality
Černý, Jakub ; Loebl, Martin (advisor) ; Hladík, Milan (referee)
This thesis formalizes a model of bounded rationality in extensive-form games called game-playing schemata. In this model, the strategies are repre- sented by a structure consisting of a deterministic finite automaton and two computational functions. The automaton represents a structured memory of the player, while the functions represent the ability of the player to identify efficient abstractions of the game. Together, the schema is a realization of a pure strategy which can be implemented by a player in order to play a given game. The thesis shows how to construct correctly playing schema for every pure strategy in any multi-player extensive-form game with perfect recall and how to evaluate its complexity. It proves that equilibria in schemata strategies always exist and computing them is PPAD-hard. Moreover, for a class of efficiently representable strategies, computing MAXPAY-EFCE can be done in polynomial time. 1
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Determinants of Interval Matrices
Matějka, Josef ; Horáček, Jaroslav (advisor) ; Hladík, Milan (referee)
This work focuses on the determinants of interval matrices. After a short introduction into interval arithmetics, the works focus on time complexity of computation tight enclosures of interval determinants, we show what complexity class this problem belongs to and how hard is approximation with relative and absolute error. Next chapter works with various preconditions of a matrix, which could lead to better results. After we analyse preconditioning of matrices we show several methods for computing determinants, starting with Gauss elimination, en- ding method using Cramer's rule. We also ponder about special cases of matrices like symmetric, tridiagonal and Toeplitz. At the end we test shown methods. 1
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Evaluation of interval polynomials
Firment, Roman ; Hladík, Milan (advisor) ; Hartman, David (referee)
In this thesis, we deal with the finding of an enclosure of the range of the real and interval polynomials in one variable. There are presented functional forms of the real polynomials which we implemented in the Matlab environment that is using interval arithmetic of the toolbox INTLAB. These forms can be used to effectively evaluate an enclosure of a polynomial. In the theoretical part there is introduced a reduction that makes possible to use an arbitrary functional form computing an enclosure of a real polynomial to evaluate an enclosure of interval polynomial. A numerical comparison is also the part of this thesis. Based on its results we designed two global functions solving our problem that apply one of the forms. A user has a possibility to indirectly influence the choice of the form by non-mandatory parameter that is specifying the strategy of computation. This parameter defines speed of evaluation and the amount of overestimation of the computed interval.
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The optimal solution set of interval linear programming problems
Garajová, Elif ; Hladík, Milan (advisor) ; Zimmermann, Karel (referee)
Determining the set of all optimal solutions of a linear program with interval data is one of the main problems discussed in interval optimization. We review two methods based on duality in linear programming, which are used to approximate the optimal set. Additionally, another decomposition method based on complementary slackness is proposed. This method provides the exact description of the optimal set for problems with a fixed coefficient matrix. The second part of the thesis is focused on studying the topological and geometric properties of the optimal set. We examine sufficient conditions for closedness, boundedness, connectedness and convexity. We also prove that testing boundedness is co- NP-hard for inequality-constrained problems with free variables. Stronger results are derived for some special classes of interval linear programs, such as problems with a fixed coefficient matrix. Furthermore, we study the effect of transformations commonly used in linear programming on interval problems, which allows for a direct generalization of some results to different types of interval linear programs. Powered by TCPDF (www.tcpdf.org)
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