Techn. University, Diss., 2004--Berlin.

Zugl.: Karlsruhe, Univ., Diss., 2005

Zugl.: Zürich, Techn. Hochsch., Diss., 2008

Techn. University, Diss., 2004--Berlin.

Universiẗat, Diss., 2005--Karlsruhe.

Frankfurt (Main), University, Diss., 2008.

Chemnitz, Techn. Univ., Masterarb., 2008.

With this thesis we bring some new results and improve some existing ones in conjugate duality and some of the areas it is applied in. First we recall the way Lagrange, Fenchel and Fenchel - Lagrange dual problems to a given primal optimization problem can be obtained via perturbations and we present some connections between them. For the Fenchel - Lagrange dual problem we prove strong duality under more general conditions than known so far, while for the Fenchel duality we show that the convexity assumptions on the functions involved can be weakened without altering the conclusion. In order to prove the latter we prove also that some formulae concerning conjugate functions given so far only for convex functions hold also for almost convex, respectively nearly convex functions. After proving that the generalized geometric dual problem can be obtained via perturbations, we show that the geometric duality is a special case of the Fenchel - Lagrange duality and the strong duality can be obtained under weaker conditions than stated in the existing literature. For various problems treated in the literature via geometric duality we show that Fenchel - Lagrange duality is easier to apply, bringing moreover strong duality and optimality conditions under weaker assumptions. The results presented so far are applied also in convex composite optimization and entropy optimization. For the composed convex cone - constrained optimization problem we give strong duality and the related optimality conditions, then we apply these when showing that the formula of the conjugate of the precomposition with a proper convex K - increasing function of a K - convex function on some n - dimensional non - empty convex set X, where K is a k - dimensional non - empty closed convex cone, holds under weaker conditions than known so far. Another field were we apply these results is vector optimization, where we provide a general duality framework based on a more general scalarization that includes as special cases and improves some previous results in the literature. Concerning entropy optimization, we treat first via duality a problem having an entropy - like objective function, from which arise as special cases some problems found in the literature on entropy optimization. Finally, an application of entropy optimization into text classification is presented.

Time-delayed actions appear as an essential component of numerous systems especially in evolution processes, natural phenomena, and particular technical applications and are associated with the existence of a memory. Under common conditions, external forces or state dependent parameters modify the length of the delay with time. Consequently, an altered dynamical behavior emerges, whose characterization is compulsory for a deeper understanding of these processes. In this thesis, the well-investigated class of time-homogeneous finite-state Markov processes is utilized to establish a variation of memory length by combining a first-order Markov chain with a memoryless Markov chain of order zero. The fluctuations induce a non-stationary process, which is accomplished for two special cases: a periodic and a random selection of the available Markov chains. For both cases, the Kolmogorov-Sinai entropy as a characteristic property is deduced analytically and compared to numerical approximations to the entropy rate of related symbolic dynamics. The convergences of per-symbol and conditional entropies are examined in order to recognize their behavior when identifying unknown processes. Additionally, the connection from Markov processes with varying memory length to hidden Markov models is illustrated enabling further analysis. Hence, the Kolmogorov-Sinai entropy of hidden Markov chains is calculated by means of Blackwell’s entropy rate involving Blackwell’s measure. These results are used to verify the previous computations.

Time-delayed actions appear as an essential component of numerous systems especially in evolution processes, natural phenomena, and particular technical applications and are associated with the existence of a memory. Under common conditions, external forces or state dependent parameters modify the length of the delay with time. Consequently, an altered dynamical behavior emerges, whose characterization is compulsory for a deeper understanding of these processes. In this thesis, the well-investigated class of time-homogeneous finite-state Markov processes is utilized to establish a variation of memory length by combining a first-order Markov chain with a memoryless Markov chain of order zero. The fluctuations induce a non-stationary process, which is accomplished for two special cases: a periodic and a random selection of the available Markov chains. For both cases, the Kolmogorov-Sinai entropy as a characteristic property is deduced analytically and compared to numerical approximations to the entropy rate of related symbolic dynamics. The convergences of per-symbol and conditional entropies are examined in order to recognize their behavior when identifying unknown processes. Additionally, the connection from Markov processes with varying memory length to hidden Markov models is illustrated enabling further analysis. Hence, the Kolmogorov-Sinai entropy of hidden Markov chains is calculated by means of Blackwell’s entropy rate involving Blackwell’s measure. These results are used to verify the previous computations.