Graduate

Thesis (Ph.D.)--University of Texas at Dallas, 2007. / Includes vita. Includes bibliographical references (leaves 105-110)

Thesis (Ph. D.)--University of New South Wales, 1999. / Also available online.

Thesis (M.S. in Operations Research)--Naval Postgraduate School, March 2009. / Thesis Advisor(s): Carlyle, W. Matthew. "March 2009." Description based on title screen as viewed on April 24, 2009. Author(s) subject terms: Integer Programming, Operational Planning, Navy Mission Planner, Navy Asset-Mission Pairing, Maritime Headquarters, Maritime Operations Center, Constrained Enumeration, Stack-based Enumeration, Mathematical Programming, Optimization, Decision Aid, Planning Tool, Ship Employment Schedule. Includes bibliographical references (p. 61-62). Also available in print.

Thesis (Ph. D. in Operations Research)--Naval Postgraduate School, Sept. 2004. / Thesis Advisor(s): R. Kevin Wood. Includes bibliographical references (p. 71-80). Also available online.

Thesis (M. Phil.)--University of Hong Kong, 1992.

Thesis (Ph. D.)--Industrial and Systems Engineering, Georgia Institute of Technology, 2006u. / Nemhauser, George L., Committee Chair ; Ahmed, Shabbir, Committee Member ; Bartholdi, John J., Committee Member ; Takriti, Samer, Committee Member ; Gu, Zonghao, Committee Member.

Many problems in economics, statistics and numerical analysis can be formulated as the optimization of a convex quadratic function over a polyhedral set. A polynomial algorithm for solving convex quadratic programming problems was first developed by Kozlov at al. (1979). Tardos (1986) was the first to present a polynomial algorithm for solving linear programming problems in which the number of arithmetic steps depends only on the size of the numbers in the constraint matrix and is independent of the size of the numbers in the right hand side and the cost coefficients. In the first part of the thesis we extended Tardos' results to strictly convex quadratic programming of the form max {cTx-½xTDx : Ax ≤ b, x ≥0} with D being symmetric positive definite matrix. In our algorithm the number of arithmetic steps is independent of c and b but depends on the size of the entries of the matrices A and D. Another part of the thesis is concerned with proximity and sensitivity of integer and mixed-integer quadratic programs. We have shown that for any optimal solution z̅ for a given separable quadratic integer programming problem there exist an optimal solution x̅ for its continuous relaxation such that / z̅ - x̅ / ∞≤n∆(A) where n is the number of variables and ∆(A) is the largest absolute sub-determinant of the integer constraint matrix A . We have further shown that for any feasible solution z, which is not optimal for the separable quadratic integer programming problem, there exists a feasible solution z̅ having greater objective function value and with / z - z̅ / ∞≤n∆(A). Under some additional assumptions the distance between a pair of optimal solutions to the integer quadratic programming problem with right hand side vectors b and b', respectively, depends linearly on / b — b' / ₁. The extension to the mixed-integer nonseparable quadratic case is also given. Some sensitivity analysis results for nonlinear integer programming problems are given. We assume that the nonlinear 0 — 1 problem was solved by implicit enumeration and that some small changes have been made in the right hand side or objective function coefficients. We then established what additional information to keep in the implicit enumeration tree, when solving the original problem, in order to provide us with bounds on the optimal value of a perturbed problem. Also, suppose that after solving the original problem to optimality the problem was enlarged by introducing a new 0 — 1 variable, say xn+1. We determined a lower bound on the added objective function coefficients for which the new integer variable xn+1 remains at zero level in the optimal solution for the modified integer nonlinear program. We discuss the extensions to the mixed-integer case as well as to the case when integer variables are not restricted to be 0 or 1. The computational results for an example with quadratic objective function, linear constraints and 0—1 variables are provided. Finally, we have shown how to replace the objective function of a quadratic program with 0—1 variables ( by an integer objective function whose size is polynomially bounded by the number of variables) without changing the set of optimal solutions. This was done by making use of the algorithm given by Frank and Tardos (1985) which in turn uses the simultaneous approximation algorithm of Lenstra, Lenstra and Lovász (1982). / Business, Sauder School of / Graduate

A dissertation submitted to the Faculty of Science School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg. April 27, 2013. / In a number of situations the derivative of the objective function of an optimization problem is not available. This thesis presents a novel algorithm for solving mixed integer programs when this is the case. The algorithm is the first developed for problems of this type which uses a trust region methodology. Three implementations of the algorithm are developed and deterministic proofs of convergence to local minima are provided for two of the implementations. In the development of the algorithm several other contributions are made. The derivative free algorithm requires the solution of several mixed integer quadratic programming subproblems and novel methods for solving nonconvex instances of these problems are developed in this thesis. Additionally, it is shown that the current definitions of local minima for mixed integer programs are deficient and a rigorous approach to developing possible definitions is proposed. Using this approach we propose a new definition which improves on those currently used in the literature. Other components of this thesis are an overview of derivative based mixed integer non-linear programming, extensive reviews of mixed integer quadratic programming and deterministic derivative free optimization and extensive computational results illustrating the effectiveness of the contributions mentioned in the previous paragraphs.

Wang, Chongyu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 74-78). / Abstract also in Chinese. / Abstract --- p.1 / Abstract in Chinese --- p.3 / Acknowledgement --- p.4 / Contents --- p.5 / List of Tables --- p.7 / List of Figures --- p.8 / Chapter 1. --- Introduction --- p.9 / Chapter 2. --- Conventional Dynamic Programming --- p.15 / Chapter 2.1. --- Principle of optimality and decomposition --- p.15 / Chapter 2.2. --- Backward dynamic programming --- p.17 / Chapter 2.3. --- Forward dynamic programming --- p.20 / Chapter 2.4. --- Curse of dimensionality --- p.23 / Chapter 3. --- Surrogate Constraint Formulation --- p.26 / Chapter 3.1. --- Surrogate constraint formulation --- p.26 / Chapter 3.2. --- Singly constrained dynamic programming --- p.28 / Chapter 3.3. --- Surrogate dual search --- p.29 / Chapter 4. --- Distance Confined Path Algorithm --- p.34 / Chapter 4.1. --- Yen´ةs algorithm for the kth shortest path problem --- p.35 / Chapter 4.2. --- Application of Yen´ةs method to integer programming --- p.36 / Chapter 4.3. --- Distance confined path problem --- p.42 / Chapter 4.4. --- Application of distance confined path formulation to integer programming --- p.50 / Chapter 5. --- Convergent Surrogate Dual Search --- p.59 / Chapter 5.1. --- Algorithm for convergent surrogate dual search --- p.62 / Chapter 5.2. --- "Solution schemes for (Pμ{αk,αβ)) and f(x) = αk" --- p.63 / Chapter 5.3. --- Computational Results and Analysis --- p.68 / Chapter 6. --- Conclusions --- p.72 / Bibliography --- p.74