This thesis proposes a new hybrid particle swarm optimizer, which employs landscape estimation and the cooperative behavior of different particles to significantly improve the performance of the original algorithm. The landscape estimation is to explore the landscape of the function in order to predict whether the function is unimodal or multimodal. Then we can decide how to optimize the function accordingly. The cooperative behavior is achieved by using two swarms, in which one swarm explores only a single dimension at a time, and the other explores all dimensions simultaneously. Furthermore, we also propose a movement tracking-based strategy to adjust the maximal velocity of the particles. This strategy can control the exploration and exploitation abilities of the swarm efficiency. Finally, we testify the performance of the proposed approach on a suite of unimodal/multimodal benchmark functions and provide comparisons with other recent variants of the PSO. The results show that our approach outperforms other methods in most of the benchmark problems.

For every positive, decreasing, summable sequence $a=(a_i)$, we can construct a Cantor set $C_a$ associated with $a$. These Cantor sets are not necessarily self-similar. Their dimensional properties and measures have been studied in terms of the sequence $a$. In this thesis, we extend these results to a more general collection of Cantor sets. We study their Hausdorff and packing measures, and compare the size of Cantor sets with the more refined notion of dimension partitions. The properties of these Cantor sets in relation to the collection of cut-out sets are then considered. The multifractal spectrum of $\mathbf{p}$-Cantor measures on these Cantor sets are also computed. We then focus on the special case of homogeneous Cantor sets and obtain a more accurate estimate of their exact measures. Finally, we prove the $L^p$-improving property of the $\mathbf{p}$-Cantor measure on a homogeneous Cantor set as a convolution operator.