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This thesis presents a series of integrated computation-orientated methods using mathematical programming (MP) approaches to carry out, in the presence of simultaneous material and geometric nonlinearities, the realistic analysis of skeletal structures that exhibit softening and limited ductility. In particular, four approaches are developed. First, the entire structural behavior is traced by using the nonholo-nomic (path-dependent) elastoplastic analysis. Second, the stepwise holonomic anal-ysis approximates the actual nonholonomic behavior by using a series of holonomic counterparts. Third, the more tractable holonomic (path-independent) analysis is implemented to approximate the overall nonholonomic response. Finally, classical limit analysis is extended to cater for this class structures; the aim is to compute in a single step ultimate load and corresponding deformations, simultaneously. The nonholonomic, stepwise holonomic and holonomic state formulations are developed as special instances of the well-known MP problem known as a mixed complementarity problem (MCP). Geometric nonlinearity is tackled via two alternative approaches, namely one that can cater for arbitrarily large deformations and the second for 2nd-order geometry effects only. The effects of combined bending and axial forces are included through a (hexagonal) piecewise linear yield locus that can accommodate either perfect plasticity or isotropic softening or hardening. The extended limit analysis problem is formulated as an instance of the challenging class of so-called mathematical programs with equilibrium constraints (MPECs). Two classes of solution approaches, namely nonlinear programming (NLP) based approaches and an equation based smoothing approach, are proposed to solve the MPEC. A number of numerical examples are provided to validate the robustness and efficiency of all proposed methods, and to illustrate some key mechanical features expected of realistic frames that exhibit local softening behavior.

Thesis (M. Phil.)--University of Hong Kong, 2008. / Also available in print.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1982. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 97-101).

Covering relaxation algorithms were first developed by Granot et al for solving positive 0-1 polynomial programming (PP) problems which maximize a linear objective function in 0-1 variables subject to a set of polynomial inequalities containing only positive coefficients ["Covering Relaxation for Positive 0-1 Polynomial Programs", Management Science, Vol. 25, (1979)]. The covering relaxation approach appears to cope successfully with the non-linearity of the PP problem and is able to solve modest size (40 variables and 40 constraints) sparse PP problems. This thesis develops a more sophisticated covering relaxation method which accelerates the performance of this approach, especially when solving PP problems with many terms in a constraint. Both the original covering relaxation algorithm and the newly introduced accelerated algorithm are cutting plane algorithms in which the relaxed problem is the set covering problem and the cutting planes are linear covering constraints. In contrast with other cutting plane methods in integer programming, the accelerated covering relaxation algorithm developed in this thesis does not solve the relaxed problem to optimality after the introduction of the cutting plane constraints. Rather, the augmented relaxed problem is first solved approximately and only if the approximate solution is feasible to the PP problem is the relaxed problem solved to optimality. The promise of this approach stems from the excellent performance of approximate procedures for solving integer programming problems. Indeed, the extensive computational experiments that were performed show that the accelerated algorithm has reduced both the number of set covering problems to be solved and the overall time required to solve a PP problem. The improvements are particularly significant for PP problems with many terms in a constraint. / Science, Faculty of / Computer Science, Department of / Graduate

Photocopy of typescript. / Digitized by Kansas State University Libraries

The basis handling procedures of the simplex method are formulated in terms of a "double basis". That is, the basis is factored as (DIAGRAM OMITTED...PLEASE SEE DAI) where ‘B, the pseudobasis matrix, is the basis matrix at the last refactorization. P and Q are permutation matrices. Forward and backward transformations and update are presented for each of two implementations of the double-basis method. The first implementation utilizes an explicit G⁻¹ matrix. The second uses a sparse LU factorization of G. Both are based on Marsten's modularized XMP package, in which standard simplex method routines are replaced by corresponding double-basis method routines. XMP and the LU double-basis method implementation employ Reid's LA05 routines for handling sparse linear programming bases. All calculations are done without reference to the H matrix. Therefore, the update is restricted to G, which has dimension limited by the refactorization frequency, and P and Q, which are held as lists. This can lead to a saving in storage space and updating time. The cost is that time for transformations will be about double. Computational comparisons of storage and speed performance are made with the standard simplex method on problems of up to 1480 constraints. It is found that, generally, the double-basis method performs best on larger, denser problems. Density seems to be the more important factor, and the problems with large nonzero growth between refactorizations are the better ones for the double-basis method. Storage saving in the basis inverse representation versus the standard method is as high as 36%, whereas the double-basis run times are 1.2 or more times as great.

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.

by Shiu-Tung Ng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 100-103). / Abstract also in Chinese. / Chapter 1 --- Preliminary --- p.5 / Chapter 1.1 --- Linear and Convex Optimization Model --- p.5 / Chapter 1.2 --- Notations for Linear Optimization --- p.5 / Chapter 1.3 --- Definition and Properties of Convexities --- p.7 / Chapter 1.4 --- Useful Theorem for Unconstrained Minimization --- p.10 / Chapter 2 --- Linear Optimization --- p.11 / Chapter 2.1 --- Self-dual Linear Optimization Model --- p.11 / Chapter 2.2 --- Definitions and Main Theorems --- p.14 / Chapter 2.3 --- Self-dual Embedding and Simple Example --- p.22 / Chapter 2.4 --- Newton step --- p.25 / Chapter 2.5 --- "Rescaling and Definition of δ(xs,w)" --- p.29 / Chapter 2.6 --- An Interior Point Method --- p.32 / Chapter 2.6.1 --- Algorithm with Full Newton Steps --- p.33 / Chapter 2.6.2 --- Iteration Bound --- p.33 / Chapter 2.7 --- Background and Rounding Procedure for Interior-point Solution --- p.36 / Chapter 2.8 --- Solving Some LP problems --- p.42 / Chapter 2.9 --- Remarks --- p.51 / Chapter 3 --- Convex Optimization --- p.53 / Chapter 3.1 --- Introduction --- p.53 / Chapter 3.1.1 --- Convex Optimization Problem --- p.53 / Chapter 3.1.2 --- Idea of Interior Point Method --- p.55 / Chapter 3.2 --- Logarithmic Barrier Method --- p.55 / Chapter 3.2.1 --- Basic Concepts and Properties --- p.55 / Chapter 3.2.2 --- k-Self-Concordance Condition --- p.62 / Chapter 3.2.3 --- Short-step Logarithmic Barrier Algorithm --- p.64 / Chapter 3.2.4 --- Initialization Algorithm --- p.67 / Chapter 3.3 --- Center Method --- p.70 / Chapter 3.3.1 --- Basic Concepts and Properties --- p.70 / Chapter 3.3.2 --- Short-step Center Algorithm --- p.75 / Chapter 3.3.3 --- Initialization Algorithm --- p.76 / Chapter 3.4 --- Properties and Examples on Self-Concordance --- p.78 / Chapter 3.5 --- Examples of Convex Optimization Problem --- p.82 / Chapter 3.5.1 --- Self-concordant Logarithmic Barrier and Distance Function --- p.82 / Chapter 3.5.2 --- General Convex Optimization Problems --- p.91 / Chapter 3.6 --- Remarks --- p.98 / Bibliography