This thesis contains an account of several modifications to two algorithms for unconstrained optimization, both of which are due to Gill and Murray. Chapter One contains a brief survey of unconstrained optimization and contains also some results which are used subsequently. Chapter Two contains an account of some work on iterative procedures for the solution of operator equations in Banach spaces due to Wolfe (1978a) in which it is suggested that it may be possible, in certain circumstances, to use high-order iterative procedures rather than Newton's method, thereby obtaining computational advantages. In Chapter Three the Newton-type algorithm of Gill and Murray (1974) is described and the ideas contained in Chapter Two are used to construct some modifications of this algorithm. Chapter Four contains some algorithms for the numerical estimation of both full and b and-type Hessian matrices. These algorithms may be used in conjunction with the optimization algorithms which are described in Chapters Three and Five. In Chapter Five the least-squares algorithm of Gill and Murray (1976) is described and the ideas contained in Chapter Two are used to construct some modifications of this algorithm. Chapter Six contains the computational results which were obtained by using the algorithms which are described in Chapters Three, Four and Five to solve the test problems which are listed in Appendices One and Two.

This thesis contains a description of algorithm, MW, for bounding the global minimizers and globally minimum value of a twice continuously differentiable function f :Rn → R1 R1 in a compact sub-interval of Rn. The algorithm MW is similar to the algorithm of Hansen (Han-80a] in that interval arithmetic is used together with certain of Hansen's ideas, but is different from Hansen's algorithm in that MW bounds the Kuhn Tucker points corresponding to the global minimizers of f in the given sab-interval. The Kuhn Tucker points are bounded with prescribed precision by using either of the algorithms KMSW [SheW-85c] or MAP [SheW-85b]. Numerical results which are obtained from Triplex [BaCM-82a] [MorC-83a] implementations of H and MW axe presented.

Dissertation submitted in fulfilment of the requirements for the degree Master of Technology: Mechanical Engineering in the Faculty of Engineering at the Cape Peninsula University of Technology 2014 / The research described in this dissertation investigates the feasibility of casting inlet valves for an internal combustion engine using Ti6Al4V alloy. The engine valves operate in an extreme environment under high thermal cycles – this requires a material that can withstand such exposures. Ti6Al4V is the most common titanium alloy with high temperature creep and fatigue resistant behaviour, however, it is not all positive. Ti6Al4V alloy also yields many difficulties with respect to processing especially when the material is cast. It is therefore important to gain a thorough understanding of the pouring and solidification characteristics of this material. The main focus of this work was to investigate and optimise feeding and geometrical parameters to produce valves that are free from defects, especially porosity. An in depth analyses of the parameters that influenced the casting quality was performed, and it was found that casting orientation, inlet feeder geometry, initial and boundary conditions all played a vital role in the final results. These parameters were individually investigated by performing detailed numerical simulations using leading simulation software for each of these cases. For each case, a minimum of ten simulations was performed to accurately determine the effect of the alteration on casting soundness and quality. Furthermore, the relationships (if any) were observed and used in subsequent optimised simulations of an entire investment casting tree. The change of geometric orientation and inlet feeder diameter and angle showed distinct relationships with occurrence of porosity. On the other hand, alteration in the pouring parameters, such as temperature and time, had negligible effect on occurrence or position of porosity in the valve. It was found that investigating individual parameters of simple geometry and then utilising these best-fit results in complex geometry yielded beneficial results that would otherwise not be attainable.

In the literature, error bound conditions have been widely used for studying the linear convergence rates of various first-order algorithms and the majority of literature focuses on how to sufficiently ensure these error bound conditions, usually posing more assumptions on the model under discussion. In this thesis, we focus on the alternating direction method of multipliers (ADMM), and show that the known error bound conditions for studying ADMM's linear convergence, can indeed be further weakened if the error bound is studied over the specific iterative sequence generated by ADMM. A so-called partial error bound condition, which is tailored for the specific ADMM's iterative scheme and weaker than known error bound conditions in the literature, is thus proposed to derive the linear convergence of ADMM. We further show that this partial error bound condition theoretically justifies the difference if the two primal variables are updated in different orders in implementing ADMM, which had been empirically observed in the literature yet no theory is known so far.

University of Macau / Faculty of Science and Technology / Department of Electromechanical Engineering

No description available.

This thesis deals with Attack and Defence Models involving four different pay-off functions In each model, by means of standard min-max and convexity arguments, the optimal attack strategy, optimal defence strategy, and value are calculated. / Science, Faculty of / Mathematics, Department of / Graduate

Analog computer methods are developed for iterative parameter optimization and continuous identification of linear systems with time delay. A unified treatment of structural sensitivity analysis is presented. New results are presented on the simultaneous generation of the second-order sensitivity functions for a class of systems. To test the theory, a simple controller is proposed for a linear time-invariant process with time delay. The controller parameters are to be adjusted so that the process output closely approximates some desired output, while remaining insensitive to fluctuations in the plant delay. For this purpose, an iterative procedure is used to minimize a combined error-sensitivity performance index. Two continuous identification methods are presented. The first of these is an output error method derived using structural sensitivity analysis, while the second is an equation error method. It is assumed that processes to be identified can be represented by linear differential-difference equations. Unknown parameters in these equations are determined by a steepest descent technique. Both methods are applied for the first time to the important problem, of identifying constant and time varying time delays. The two methods are compared, and the equation error method is found to be superior. It is shown that this method is stable in the linear region in parameter space, and is always stable if all process delays are known. Techniques leading to a more economical implementation of the equation error method are presented. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

This thesis is concerned with estimation and control of linear distributed parameter systems. For the estimation of linear deterministic continuous-time distributed parameter systems, a linear deterministic distributed parameter filter that yields the state estimate based on noiseless linear measurements available over the complete occupied spatial domain is derived by consideration of a Lyapunov type of stability. The general results are then specialized to the case when noiseless linear measurements are available at only several points in the spatial domain. A numerical example illustrates its use in an overall control scheme. For the estimation of linear stochastic discrete-time distributed parameter systems, a linear discrete-time distributed parameter filter having a predictor-corrector structure, that yields the minimum-variance estimate of the state based on noise-corrupted linear measurements assumed available at only several spatial locations, is derived. The filtered estimate and the filtering error are shown to satisfy an orthogonal projection lemma, whence a Wiener-Hopf equation is derived. The filter is implementable on-line and a numerical example illustrates its use. The optimal pointwise regulation control problem for linear stochastic discrete-time distributed parameter systems is treated through application of dynamic programming. The separation of the complete control scheme into the estimation and control subsystems is shown. Its usefulness is illustrated in a numerical example. By first expanding Green's function and then considering the limiting behaviour of the corresponding discrete-time results on estimation and control obtained previously, solutions of the continuous-time linear minimum-variance filtering estimation and optimal pointwise regulation control problems for linear stochastic continuous-time distributed parameter systems are obtained. Further, a separation theorem is obtained and Kalman's duality theorem extended. For the pointwise regulation control problem of linear stochastic discrete-time distributed parameter systems, the case of unknown noise characteristics is treated. Based on an examination of the open-loop-optimal feedback control approach, a suboptimal control scheme is proposed. A filter that is adaptively selected on-line based on minimizing an instantaneous cost functional so derived from the original one as to realize a trade-off between control and estimation costs is put forward. A numerical example shows the effectiveness of the suboptimal control scheme in comparison with the optimal one. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

M.Sc. (Computer Science) / Chapter 1 is a summary in which the problems- discussed in this study, as well as the relationship between them are shown. The algorithmic notation used when discussing the problems are also defined. In chapter 2 the three classes of algorithms for finding a minimum spanning tree, i.e. the algorithms of Prim, Kruskal and SolIin are discussed. PASCAL implementations of all the algorithms are presented. We also report on our computational experience with these implementations. It was found that the implementation of the Prim algorithm was very efficient, while implementations of the Kruskal algorithm also gave good results. Implementations of the Sollin algorithm were less efficient, because of the complex data structures involved. Algorithms for finding an optimum arborescence or branching in a network have independently been proposed by Edmonds, Chu & Liu as well as Bock. Tarjan as well as Camerini et al subsequently discussed aspects of the efficient implementation of the algorithm. In chapter 3 we draw attention to the related work of Fulkerson by reformulating the Edmonds-Fulkerson algorithm and giving a simple proof of the correctness of the algorithm. We also discuss and present an implementation of the algorithm and report on our computational experience. From the results presented it is clear that the number of cycles encountered during the first phase of the algorithm has a significant effect on the efficiency of the algorithm...