Convergence rates of Markov chains have been widely studied in recent years. In particular, quantitative bounds on convergence rates have been studied in various forms by Meyn and Tweedie [Ann. Appl.

Simulated annealing is a popular and much studied method for maximizing functions on finite or compact spaces. For noncompact state spaces, the method is still sound, but convergence results are scarce. We show here how to prove convergence in such cases, for Markov chains satisfying suitable drift and minorization conditions.

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In this paper simulated annealing algorithms for continuous global optimization are considered. Under the simplifying assumption of known optimal value, the convergence of the algorithms and an upper bound for the expected first hitting time, i.e. the expected number of iterations before reaching the global optimum value within accuracy ", are established. The obtained results are compared with those for the ideal algorithm PAS (Pure Adaptive Search) and for the simple PRS (Pure Random Search) algorithm. KEYWORDS: global optimization, simulated annealing, convergence, first hitting time 1 Introduction The simulated annealing approach was inspired by a physical phenomenon. If we reduce the temperature of a liquid, the thermal mobility of the molecules is lost. If the decrease is slow enough a pure crystal is formed, corresponding to a state of minimum energy. If the decrease is too fast a polycrystalline or an amorphous state with higher energy are reached. In [21] a Monte Carlo meth...

Abstract. We define a new method for global optimization, the Homotopy Optimization Method (HOM). This method differs from previous homotopy and continuation methods in that its aim is to find a minimizer for each of a set of values of the homotopy parameter, rather than to follow a path of minimizers. We define a second method, called HOPE, by allowing HOM to follow an ensemble of points obtained by perturbation of previous ones. We relate this new method to standard methods such as simulated annealing and show under what circumstances it is superior. We present results of extensive numerical experiments demonstrating performance of HOM and HOPE.

I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research.

Automated design of superscalar processors can provide future system-on-chip (SOC) designers with a key-turn method of generating superscalar processors that are Pareto-optimal in terms of performance, energy consumption, and area for the target application program(s). Unfortunately, current optimization methods are based on time-consuming cycle-accurate simulation, unsuitable for analysis of hundreds of thousands of design options that is required to arrive at Pareto-optimal designs. This dissertation bridges the gap between a large design space of superscalar processors and the inability of cycle-accurate simulation to analyze a large design space, by providing a computationally and conceptually simple analytical method for generating Pareto-optimal superscalar processor designs. The proposed and evaluated analytical method consists of three parts: (1) a method for analytically estimating the performance in terms a cycles-per-instruction (CPI) using the application program statistics and the superscalar processor parameters, (2) a method of analytically estimating various energy consuming activities using the application program statistics and the superscalar processor parameters, and (3) a method of finding the Pareto-

Summary. The prediction of protein native conformations is still a big challenge in science, although a strong research activity has been carried out on this topic in the last decades. In this chapter we focus on ab-initio computational methods for protein fold predictions that do not rely heavily on comparisons with known protein structures and hence appear to be the most promising methods for determining conformations not yet been observed experimentally. To identify main trends in the research concerning protein fold predictions, we briefly review several ab-initio methods, including a recent topological approach that models the protein conformation as a tube having maximum thickness without any self-contacts. This representation leads to a constrained global optimization problem. We introduce a modification in the tube model to increase the compactness of the computed conformations, and present results of computational experiments devoted to simulating α-helices and all-α proteins. A Metropolis Monte Carlo Simulated Annealing algorithm is used to search the protein conformational space.

We illustrate our experience in developing and implementing algorithms for map merging, i.e., the problem of fusing two or more partial maps without common reference frames into one large global map. The partial maps may for example be acquired by multiple robots, or during several runs of a single robot from varying starting positions. Our work deals with low quality maps based on probabilistic grids, motivated by the goal to develop multiple mobile platforms to be used in rescue environments. Several contributions to map merging are presented. First of all, we address map merging using a motion planning algorithm. The merging process can be done by rotating and translating the partial maps until similar regions overlap. Second, a motion planning algorithm is presented which is particular suited for this task. Third, a special metric is presented which guides the motion planning algorithm towards the goal of optimally overlapping partial maps. Results with our approach are presented based on data gathered from real robots developed for the RoboCupRescue real robot league.

This thesis concerns numerical methods for mapping of multiple quantitative trait loci, QTL. Interactions between multiple genetic loci influencing important traits, such as growth rate in farm animals and predisposition to cancer in humans, make it necessary to search for several QTL simultaneously. Simultaneous search for n QTL involves solving an n-dimensional global optimization problem, where each evaluation of the objective function consists of solving a generalized least squares problem. In Paper A we present efficient algorithms, mainly based on updated QR factorizations, for evaluating the objective functions of different parametric QTL mapping methods. One of these algorithms reduces the computational work required for an important function class by one order of magnitude compared with the best of the methods used by other authors. In Paper B previously utilized techniques for finding the global optimum of the objective function are compared with a new approach based on the DIRECT algorithm of Jones et al. The new method gives accurate results in one order of magnitude less time than the best of the formerly employed algorithms. Using the algorithms presented in Papers A and B, simultaneous search for at least three QTL, including computation of the relevant empirical significance thresholds, can be performed routinely.