Continuous Approach To Discrete Optimum Design

In terms of the development of computer technology, finite element method and mathematical programming, the optimum design of structures has experienced fast growth both in the theory and techniques, and become necessary tools in modern design. At present, continuous optimum design has been developed successfully. However, the development of discrete optimum design is not very satisfactory. Classical methods and recent arisen stochastic search methods can only solve some small-scale problems. Some complex algorithms (interior point method and semidefinite programming method) have the capability of sovling relative large-scale problems, but they are hardly to be mastered by the engineers. The aim of this thesis is to explore a new approach to discrete optimum design, and the objective is to sovle large-scale disrete optimization problems by using available general nonlinear optimization software. Although the dissertation deals with some mathematical theory, the emphasis of this thesis is put on the aspect of numerical implementation. Finally, continuous algorithms are applied to some structural optimization problems, to verify the practical applicability of the proposed approach.Main contents of this dissertation are organized as follows:In chapter 1, the background of this dissertation is introduced and some existing methods for solving 0-1 programming, discrete optimum design, and topology optimization are surveyed. Main research areas and structure of this dissertation are discussed.In chapter 2, three kinds of continuous approaches to 0-1 programming are investigated: sigmoid function approach, NCP function approach and binary entropy function approach. With these functions, the original problem is converted to an equivalent continuous optimization problem, and becomes solvable by using available nonlinear optimization software.In chapter 3, we focus on the binary quadratic problem which is a typical problem in combinatorial optimization, and perform further analysis for the NCP function approach. With NCP function, the original BQP problem is converted to a completely equivalent continuous optimization problem. An augmented Lagrangian method is presented. The augmented Lagrangian function is conditional convex in a larger region for sufficiently large penalty parameters, which makes it possible that a global solution of the original problem could be obtained. To validate the applicability, robustness and solution quality of proposed approach,...