Mathematical optimization

In this dissertation we will present a stochastic optimization algorithm and use it and other mathematical techniques to tackle problems arising in medicinal chemistry. In Chapter 1, we present some background about stochastic optimization and the Accelerated Random Search (ARS) algorithm. We then present a novel improvement of the ARS algorithm, DIrected Accelerated Random Search (DARS), motivated by some theoretical results, and demonstrate through numerical results that it improves upon ARS. In Chapter 2, we use DARS and other methods to address issues arising from the use of mixture-based combinatorial libraries in drug discovery. In particular, we look at models associated with the biological activity of these mixtures and use them to answer questions about sensitivity and robustness, and also present a novel method for determining the integrity of the synthesis. Finally, in Chapter 3 we present an in-depth analysis of some statistical and mathematical techniques in combinatorial chemistry, including a novel probabilistic approach to using structural similarity to predict the activity landscape.

The use of permutations in data compression is an aspect that is worthy of further exploration. The work that has been done in video compression based on permutations was primarily oriented towards lossless algorithms. The study of previous algorithms has led to a new algorithm that could be either lossless or lossy, for which the amount of compression and the quality of the output can be controlled. The lossless version of our algorithm performs close to lossy versions of H.264 and it improves on them for the majority of the videos that we analyzed. Our algorithm could be used in situations where there is a need for lossless compression and the video sequences are part of a single scene, e.g., medical videos, where loss of information could be risky or expensive. Some results on permutations, which may be of independent interest, arose in developing this algorithm. We report on these as well.