Global Algorithms for Nonlinear Discrete Optimization and Discrete-Valued Optimal Control Problems

Optimal control problems arise in many applications, such as in economics, finance, process engineering, and robotics . Some optimal control problems involve a control which takes values from a discrete set. These problems are known as discrete-valued optimal control problems. Most practical discre...

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Main Author:	Woon, Siew Fang
Format:	Thesis
Language:	eng
Published:	2009
Subjects:	QA273-280 Probabilities Mathematical statistics
Online Access:	https://etd.uum.edu.my/3537/1/2319.pdf
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id	my-uum-etd.3537
record_format	uketd_dc
institution	Universiti Utara Malaysia
collection	UUM ETD
language	eng
advisor	Rehbock, Volker
topic	QA273-280 Probabilities Mathematical statistics
spellingShingle	QA273-280 Probabilities Mathematical statistics Woon, Siew Fang Global Algorithms for Nonlinear Discrete Optimization and Discrete-Valued Optimal Control Problems
description	Optimal control problems arise in many applications, such as in economics, finance, process engineering, and robotics . Some optimal control problems involve a control which takes values from a discrete set. These problems are known as discrete-valued optimal control problems. Most practical discrete-valued optimal control problems have multiple local minima and thus require global optimization methods to generate practically useful solutions. Due to the high complexity of these problems, metaheuristic based global optimization techniques are usually required. One of the more recent global optimization tools in the area of discrete optimization is known as the discrete filled function method. The basic idea of the discrete filled function method is as follows. We choose an initial point and then perform a local search to find an initial local minimizer. Then, we construct an auxiliary function, called a discrete filled function, at this local minimizer. By minimizing the filled function, either an improved local minimizer is found or one of the vertices of the constraint set is reached. Otherwise, the parameters of the filled function are adjusted. This process is repeated until no better local minimizer of the corresponding filled function is found. The final local minimizer is then taken as an approximation of the global minimizer. While the main aim of this thesis is to present a new computational methodfor solving discrete-valued optimal control problems, the initial focus is on solvingpurely discrete optimization problems. We identify several discrete filled functionstechniques in the literature and perform a critical review including comprehensive numerical tests. Once the best filled function method is identified, we propose and test several variations of the method with numerical examples. We then consider the task of determining near globally optimal solutions of discrete-valued optimal control problems. The main difficulty in solving the discrete-valued optimal control problems is that the control restraint set is discrete and hence not convex. Conventional computational optimal control techniques are designed for problems in which the control takes values in a connected set, such as an interval, and thus they cannot solve the problem directly. Furthermore, variable switching times are known to cause problems in the implementation of any numerical algorithm due to the variable location of discontinuities in the dynamics. Therefore, such problem cannot be solved using conventional computational approaches. We propose a time scaling transformation to overcome this difficulty, where a new discrete variable representing the switching sequence and a new variable controlling the switching times are introduced. The transformation results in an equivalent mixed discrete optimization problem. The transformed problemis then decomposed into a bi-level optimization problem, which is solved using a combination of an efficient discrete filled function method identified earlier and a computational optimal control technique based on the concept of control parameterization. To demonstrate the applicability of the proposed method, we solve two complex applied engineering problems involving a hybrid power system and a sensor scheduling task, respectively. Computational results indicate that this method is robust, reliable, and efficient. It can successfully identify a near-global solution for these complex applied optimization problems, despite the demonstrated presence of multiple local optima. In addition, we also compare the results obtained with other methods in the literature. Numerical results confirm that the proposed method yields significant improvements over those obtained by other methods.
format	Thesis
qualification_name	Ph.D.
qualification_level	Doctorate
author	Woon, Siew Fang
author_facet	Woon, Siew Fang
author_sort	Woon, Siew Fang
title	Global Algorithms for Nonlinear Discrete Optimization and Discrete-Valued Optimal Control Problems
title_short	Global Algorithms for Nonlinear Discrete Optimization and Discrete-Valued Optimal Control Problems
title_full	Global Algorithms for Nonlinear Discrete Optimization and Discrete-Valued Optimal Control Problems
title_fullStr	Global Algorithms for Nonlinear Discrete Optimization and Discrete-Valued Optimal Control Problems
title_full_unstemmed	Global Algorithms for Nonlinear Discrete Optimization and Discrete-Valued Optimal Control Problems
title_sort	global algorithms for nonlinear discrete optimization and discrete-valued optimal control problems
granting_institution	Curtin University of Technology
granting_department	College of Arts and Sciences (CAS)
publishDate	2009
url	https://etd.uum.edu.my/3537/1/2319.pdf
_version_	1747827597242269696
spelling	my-uum-etd.35372013-09-15T01:39:33Z Global Algorithms for Nonlinear Discrete Optimization and Discrete-Valued Optimal Control Problems 2009 Woon, Siew Fang Rehbock, Volker College of Arts and Sciences (CAS) Department of Mathematics and Statistics QA273-280 Probabilities. Mathematical statistics Optimal control problems arise in many applications, such as in economics, finance, process engineering, and robotics . Some optimal control problems involve a control which takes values from a discrete set. These problems are known as discrete-valued optimal control problems. Most practical discrete-valued optimal control problems have multiple local minima and thus require global optimization methods to generate practically useful solutions. Due to the high complexity of these problems, metaheuristic based global optimization techniques are usually required. One of the more recent global optimization tools in the area of discrete optimization is known as the discrete filled function method. The basic idea of the discrete filled function method is as follows. We choose an initial point and then perform a local search to find an initial local minimizer. Then, we construct an auxiliary function, called a discrete filled function, at this local minimizer. By minimizing the filled function, either an improved local minimizer is found or one of the vertices of the constraint set is reached. Otherwise, the parameters of the filled function are adjusted. This process is repeated until no better local minimizer of the corresponding filled function is found. The final local minimizer is then taken as an approximation of the global minimizer. While the main aim of this thesis is to present a new computational methodfor solving discrete-valued optimal control problems, the initial focus is on solvingpurely discrete optimization problems. We identify several discrete filled functionstechniques in the literature and perform a critical review including comprehensive numerical tests. Once the best filled function method is identified, we propose and test several variations of the method with numerical examples. We then consider the task of determining near globally optimal solutions of discrete-valued optimal control problems. The main difficulty in solving the discrete-valued optimal control problems is that the control restraint set is discrete and hence not convex. Conventional computational optimal control techniques are designed for problems in which the control takes values in a connected set, such as an interval, and thus they cannot solve the problem directly. Furthermore, variable switching times are known to cause problems in the implementation of any numerical algorithm due to the variable location of discontinuities in the dynamics. Therefore, such problem cannot be solved using conventional computational approaches. We propose a time scaling transformation to overcome this difficulty, where a new discrete variable representing the switching sequence and a new variable controlling the switching times are introduced. The transformation results in an equivalent mixed discrete optimization problem. The transformed problemis then decomposed into a bi-level optimization problem, which is solved using a combination of an efficient discrete filled function method identified earlier and a computational optimal control technique based on the concept of control parameterization. To demonstrate the applicability of the proposed method, we solve two complex applied engineering problems involving a hybrid power system and a sensor scheduling task, respectively. Computational results indicate that this method is robust, reliable, and efficient. It can successfully identify a near-global solution for these complex applied optimization problems, despite the demonstrated presence of multiple local optima. In addition, we also compare the results obtained with other methods in the literature. Numerical results confirm that the proposed method yields significant improvements over those obtained by other methods. 2009 Thesis https://etd.uum.edu.my/3537/ https://etd.uum.edu.my/3537/1/2319.pdf text eng staffonly http://espace.library.curtin.edu.au/R?func=dbin-jump-full&local_base=gen01-era02&object_id=188539 Ph.D. doctoral Curtin University of Technology N. U. Ahmed. Linear and nonlinear filtering for scientists and engineers. Singapore: World Scientific, 1998. M. Ashari. Optimisation of photovoltaic/diesel/battery hybrid power systems for remote area electrification. Master’s thesis, Department of Electrical and Computer Engineering, Curtin University of Technology, 1997. J. Bang-Jensen, G. Gutin, and A. Yeo. 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