Browsing by Author "Shu-Cherng Fang, Committee Chair"

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Bivariate Cubic L1 Splines and Applications
(2007-11-06) Zhang, Wei; Shu-Cherng Fang, Committee Chair; Elmor L. Peterson, Committee Member; Henry L.W. Nuttle, Committee Member; Xiuli Chao, Committee Member; John E. Lavery, Committee Member
Bivariate cubic L1 splines can provide shape-preserving surfaces for various applications. Using the reduced Hsieh-Clough-Tocher (rHCT) elements on the triangulated irregular networks (TINs), we model a bivariate cubic L1 spline as the solution to a nonsmooth convex programming problem. This problem is a generalized geometric programming (GGP) problem, whose dual problem is to optimize a linear objective function over convex cubic constraints. Using a linear programming transformation, a dual optimal solution can be converted to a desired primal solution. For computational efficiency, we further develop a compressed primal-dual interior-point method to directly calculate an approximated primal optimal solution. This compressed primal-dual algorithm can handle terrain data over hundreds-by-hundreds grids using a personal computer. However, for real-life applications, terrain data are given in thousands-by-thousands grids. To meet the computational challenge, we establish a "non-iterative" domain decomposition principle to reduce the computational requirements. We have also conducted computational experiments to show that the proposed domain decomposition principle can handle large size data for real terrain applications.
Fuzzy Relational Equations: Resolution and Optimization
(2009-12-02) Li, Pingke; Shu-Cherng Fang, Committee Chair; Simon M. Hsiang, Committee Member; Yahya Fathi, Committee Member; James R. Wilson, Committee Member
Fuzzy relational equations play an important role as a platform in various applications of fuzzy sets and systems. The resolution and optimization of fuzzy relational equations are of our particular interests from both of the theoretical and applicational viewpoints. In this dissertation, fuzzy relational equations are treated in a unified framework and classified according to different aspects of their composite operations. For a given finite system of fuzzy relational equations with a specific composite operation, the consistency of the system can be verified in polynomial time by constructing a potential maximum/minimum solution and characteristic matrix. The solution set of a consistent system can be characterized by a unique maximum solution and finitely many minimal solutions, or dually, by a unique minimum solution and finitely many maximal solutions. The determination of all minimal/maximal solutions is closely related to the detection of all irredundant coverings of a set covering problem defined by the characteristic matrix, which may involve additional constraints. In particular, for fuzzy relational equations with sup-T composition where T is a continuous triangular norm, the existence of the additional constraints depends on whether T is Archimedean or not. Fuzzy relational equation constrained optimization problems are investigated as well in this dissertation. It is shown that the problem of minimizing an objective function subject to a system of fuzzy relational equations can be reduced in general to a 0-1 mixed integer programming problem. If the objective function is linear, or more generally, separable and monotone in each variable, then it can be further reduced to a set covering problem. Moreover, when the objective function is linear fractional, it can be reduced to a 0-1 linear fractional optimization problem and then solved via parameterization methods. However, if the objective function is max-separable with continuous monotone or unimodal components, then the problem can be solved efficiently, and its optimal solution set can be well characterized.
Game Theoretic Analysis of a Distribution System in Supply Chain
(2003-06-20) Yuan, Pei-Lun; Shu-Cherng Fang, Committee Chair; Henry L. W. Nuttle, Committee Co-Chair; Xiuli Chao, Committee Member
We consider a distribution system in which one supplier provides a single product to several retailers at the beginning of a selling season. The supplier has infinite capacity. The customer demand at each retailer is randomly distributed. Customers who encounter a stockout at one retailer may search other retailers for the product. We study the effects of this market search behavior under both decentralized and centralized control. For the decentralized control model, we show the necessary and sufficient conditions for the existence of a Nash equilibrium, and the sufficient conditions for its uniqueness. For the centralized control model, we find that the payoff function is submodular, and thus we can only obtain allocations that are locally optimal for the entire supply chain. We also design a channel coordination mechanism to match the allocations in the decentralized control model with one of the local optimal allocations under centralized control.
Interval Computations For Fuzzy Relational Equations And Cooperative Game Theory
(2003-03-06) Wang, Shunmin; Henry L. W. Nuttle, Committee Co-Chair; Jeffrey A. Joines, Committee Member; Robert E. Young, Committee Member; Shu-Cherng Fang, Committee Chair
This dissertation introduces the concepts of the tolerable solution set, united solution set, and controllable solution set of interval-valued fuzzy relational equations. Given a continuous t-norm, it is proved that each of the three types of the solution sets of interval-valued fuzzy relational equations with a max-t-norm composition, if nonempty, is composed of one maximum solution and a finite number of minimal solutions. Necessary and sufficient conditions for the existence of solutions are given. Computational procedures based on the constructive proofs are proposed to generate the complete solution sets. Examples are given to illustrate the procedures. Similarly, it is also proved that each type of solution set of interval-valued fuzzy relational equations with a min-s-norm composition, if nonempty, is composed of one minimum solution and a finite number of maximal solutions. For interval-valued games, a new method for ranking interval numbers is introduced. Interval-valued cooperative games are defined based on this method. Three axioms as desired properties of an interval-valued cooperative game were proposed. It is proved that a unique payoff function, which is similar to the Shapley value function, exists and satisfies the proposed axioms. Furthermore, this payoff function can be applied to non-superadditive games.
Min-Cost Multicommodity Network Flows: A Linear Case for the Convergence and Reoptimization of Multiple Single-Commodity Network Flows
(2009-05-11) Kramer, Jeremy Daniel; Thom J. Hodgson, Committee Member; William J. Stewart, Committee Member; Shu-Cherng Fang, Committee Chair
Network Flow problems are prevalent in Operations Research, Computer Science, Industrial Engineering and Management Science. They constitute a class of problems that are frequently faced by real world applications, including transportation, telecommunications, production planning, etc. While many problems can be modeled as Network Flows, these problems can quickly become unwieldy in size and difficult to solve. One particularly large instance is the Min-Cost Multicommodity Network Flow problem. Due to the time-sensitive nature of the industry, faster algorithms are always desired: recent advances in decomposition methods may provide a remedy. One area of improvement is the cost reoptimization of the min-cost single commodity network flow subproblems that arise from the decomposition. Since similar single commodity network flow problems are solved, information from the previous solution provides a "warm-start" of the current solution. While certain single commodity network flow algorithms may be faster "from scratch," the goal is to reduce the overall time of computation. Reoptimization is the key to this endeavor. Three single commodity network flow algorithms, namely, cost scaling, network simplex and relaxation, will be examined. They are known to reoptimize well. The overall goal is to analyze the effectiveness of this approach within one particular class of network problems.
Neural Networks for Pattern Classification and Universal Approximation
(2002-07-08) Liao, Yi; Shu-Cherng Fang, Committee Chair; Henry L. W. Nuttle, Committee Co-Chair; Yuan-Shin Lee, Committee Member; Jesus Rodriguez, Committee Member
This dissertation studies neural networks for pattern classification and universal approximation. The objective is to develope a new neural network model for pattern classification, and relax the conditions for Radial-Basis Function networks to be universal approximators. First, the problem of pattern classification is introduced, which is followed by a brief introduction of three popular nonlinear classification techniques, that is, Multi-Layer Perceptrons (MLP), Radial-Basis Function (RBF) networks, and Support Vector Machines (SVM). Then, based on the basic concepts of MLP, RBF and SVM, a new neural network model with bounded weights is proposed, and some experimental results are reported. Later, the problem of universal approximation by neural networks is introduced, and the researches on ridge activation functions and radial-basis activation functions are reviewed. Then, the relaxed conditions for RBF networks to be universal approximators are presented. We show that RVF networks can uniformly approximate any continuous function on a compact set provided that the radial basis activation function is continuous almost every where, locally essentially bounded, and not a polynomial. Some experimental results are reported to illustrate our findings. The dissertation ends with the conclusion and future research.
Tabu Search and Genetic Algorithm for Phylogeny Inference
(2008-10-21) Lin, Yu-Min; Shu-Cherng Fang, Committee Chair; Jeffrey L. Thorne, Committee Co-Chair; Henry L. W. Nuttle, Committee Member; Steffen Heber, Committee Member
Phylogenetics is the study of evolutionary relations between different organisms. Phylogenetic trees are the representations of these relations. Researchers have been working on finding fast and systematic approaches to reconstruct phylogenetic trees from observed data for over 40 years. It has been shown that, given a certain criterion to evaluate each tree, finding the best fitted phylogenetic trees among all possible trees is an NP-hard problem. In this study, we focus on the topology searching techniques for the maximum-parsimony and maximum-likelihood phylogeny inference. We proposed two search methods based on tabu search and genetic algorithms. We first explore the feasibility of using tabu search for finding the maximum-parsimony trees. The performance of the proposed algorithm is evaluated based on its efficiency and accuracy. Then we proposed a hybrid method of the tabu search and genetic algorithm. The experimental results indicate that the hybrid method can provide maximum-parsimony trees with a ggood level of accuracy and efficiency. The hybrid method is also implemented for finding maximum-likelihood trees. The experimental results show that the proposed hybrid method produce better maximum-likelihood trees than the default-setting dnaml program in average on the tested data sets. On a much larger data set, the hybrid method outperforms the default-setting dnaml program and has equally good performance as the dnaml program with the selected jumble option.
Theory and algorithms for cubic L1 splines
(2003-02-09) Cheng, Hao; Shu-Cherng Fang, Committee Chair; Henry L.W. Nuttle, Committee Co-Chair; Yahya Fathi, Committee Member; John E. Lavery, Committee Member; Elmor L. Peterson, Committee Member; Hien T. Tran, Committee Member
In modern geometric modeling, one of the requirements for interpolants is that they 'preserve shape well.' Shape preservation has often been associated with preservation of monotonicity and convexity/concavity. While shape preservation cannot yet be defined quantitatively, it is generally agreed that shape preservation involves eliminating extraneous non-physical oscillation. Classical splines, which exhibit extraneous oscillation, do not 'preserve shape well.' Recently, Lavery introduced a new class of cubic L1 splines. Empirical experiment has shown that cubic L1 splines are cable of providing C¹-smooth, shape-preserving, multi-scale interpolation of arbitrary data, including data with abrupt changes in spacing and magnitude, with no need for monotonicity or convexity constraints, node adjustment or other user input. However, the shape-preserving capability of cubic L1 splines has not been proved theoretically. The currently available algorithm only provides an approximation to the coefficients of cubic L1 splines. To lay the groundwork for theoretical analysis and the development of an exact algorithm, this dissertation proposes to treat cubic L1 spline problems in a geometric programming framework. Such a framework leads to a geometric dual problem with a linear objective function and convex quadratic constraints. It also provides a linear system for dual-to-primal conversion. We prove that cubic L1 splines preserve shape well, in particular, in eliminating non-physical oscillations, without review of raw data or any human intervention. We also show that cubic L1 splines perform well for multi-scale data, as well as preserve linearity and convexity/concavity under mild conditions. An exact algorithm based on the geometric programming model is proposed for solving cubic L1 splines. It decomposes the geometric programming problem into several independent small-sized sub-problems and applies a specialized active set algorithm to solve the sub-problems. The algorithm is numerically stable and highly parallelizable. It requires only simple algebraic operations.
Theory and Algorithms for Shape-preserving Bivariate Cubic L1 Splines.
(2005-04-12) Wang, Yong; Shu-Cherng Fang, Committee Chair
A major objective of modelling geophysical features, biological objects, financial processes and many other irregular surfaces and functions is to develop "shape-preserving" methodologies for smoothly interpolating bivariate data with sudden changes in magnitude or spacing. Shape preservation usually means the elimination of extraneous non-physical oscillations. Classical splines do not preserve shape well in this sense. Empirical experiments have shown that the recently proposed cubic L₁ splines are cable of providing C₁-smooth, shape-preserving, multi-scale interpolation of arbitrary data, including data with abrupt changes in spacing and magnitude, with no need for node adjustment or other user input. However, a theoretic treatment of the bivariate cubic L₁ splines is still lacking. The currently available approximation algorithms are not able to generate the exact coefficients of a bivariate cubic L₁ spline. For theoretical treatment and the algorithm development, we propose to solve bivariate cubic L*#8321; spline problems in a generalized geometric programming framework. Our framework includes a primal problem, a geometric dual problem with a linear objective function and convex cubic constraints, and a linear system for dual-to-primal transformation. We show that bivariate cubic L₁ splines indeed preserve linearity under some mild conditions. Since solving the dual geometric program involves heavy computation, to improve computational efficiency, we further develop three methods for generating bivariate cubic L₁ splines: a tensor-product approach that generates a good approximation for large scale bivariate cubic L₁ splines; a primal-dual interior point method that obtains discretized bivariate cubic L₁ splines robustly for small and medium size problems; and a compressed primal-dual method that efficiently and robustly generates discretized bivariate cubic L₁ splines of large size.