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Convexity in chance constraints programming
Olos, Marek ; Kopa, Miloš (advisor) ; Adam, Lukáš (referee)
1 Abstract: This thesis deals with chance constrained stochastic programming problems. We consider several chance constrained models and we focus on their convexity property. The thesis presents the theory of α-concave functions and measures as a basic tool for proving the convexity of the problems. We use the results of the theory to prove the convexity of the models first for the continu- ous distributions, then for the discrete distributions of the random vectors. We characterize a large class of the continuous distributions, that satisfy the suffi- cient conditions for the convexity of the given models and we present solving algorithms for these models. We present sufficient conditions for the convexity of the problems with dicrete distributions, too. We also deal with the algorithms for solving non-convex problems and briefly discuss the difficulties that can occur when using these methods.
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Convexity in chance constraints programming
Olos, Marek ; Kopa, Miloš (advisor) ; Adam, Lukáš (referee)
This thesis deals with chance constrained stochastic programming pro- blems. The first chapter is an introduction. We formulate several stochastic pro- gramming problems in the second chapter. In chapter 3 we present the theory of α-concave functions and measures as a basic tool for proving convexity of the problems formulated in chapter 2 for the continuous distributions of the random vectors. We use the results of the theory to characterize a large class of the conti- nuous distributions, that satisfy the sufficient conditions for the convexity and to prove convexity of concrete sets. In chapter 4 we present sufficient conditions for the convexity of the problems and we briefly discuss the method of the p-level ef- ficient points. In chapter 5 we solve a portfolio selection problem using Kataoka's model. 1
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Convexity in chance constraints programming
Olos, Marek ; Kopa, Miloš (advisor) ; Adam, Lukáš (referee)
This thesis deals with chance constrained stochastic programming pro- blems. The first chapter is an introduction. We formulate several stochastic pro- gramming problems in the second chapter. In chapter 3 we present the theory of α-concave functions and measures as a basic tool for proving convexity of the problems formulated in chapter 2 for the continuous distributions of the random vectors. We use the results of the theory to characterize a large class of the conti- nuous distributions, that satisfy the sufficient conditions for the convexity and to prove convexity of concrete sets. In chapter 4 we present sufficient conditions for the convexity of the problems and we briefly discuss the method of the p-level ef- ficient points. In chapter 5 we solve a portfolio selection problem using Kataoka's model. 1
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Convexity in chance constraints programming
Olos, Marek ; Kopa, Miloš (advisor) ; Adam, Lukáš (referee)
1 Abstract: This thesis deals with chance constrained stochastic programming problems. We consider several chance constrained models and we focus on their convexity property. The thesis presents the theory of α-concave functions and measures as a basic tool for proving the convexity of the problems. We use the results of the theory to prove the convexity of the models first for the continu- ous distributions, then for the discrete distributions of the random vectors. We characterize a large class of the continuous distributions, that satisfy the suffi- cient conditions for the convexity of the given models and we present solving algorithms for these models. We present sufficient conditions for the convexity of the problems with dicrete distributions, too. We also deal with the algorithms for solving non-convex problems and briefly discuss the difficulties that can occur when using these methods.
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