A sequential linearization approach for mixed-discrete nonlinear design optimization.

Abstract

In design optimization, variables that can only take discrete values occur naturally and frequently in the mathematical models of the design. Techniques currently used to solve mixed-discrete nonlinear design optimization problems are generally computationally expensive. This Thesis examines a sequential linearization strategy for solving mixed-discrete nonlinear programming problems. The approach is to solve a sequence of mixed-discrete linear programming problems that are approximations to the nonlinear problems. These mixed-discrete linear problems are solved using the simplex method and Dakin's branch and bound rule. By using ideas of decreasing step bounds and $\\varepsilon$-feasibility, this sequential linearization approach forces the solutions of the mixed-discrete linear approximations to converge for nonlinear problems with pseudoconvex objective functions and convex constraints, except in certain rare cases. The implementation of an algorithm based on the sequential linearization strategy is tested on numerical examples solved by researchers using other approaches. The results show that this approach requires fewer objective and constraint function evaluations than most other methods. The algorithm is further applied to some engineering design problems. These problems involve discrete variables arising from material selection and st and ard size selection. The results show that the sequential linearization approach is able to solve efficiently problems containing such discrete variables. Some of the problems solved require stress and deflection information from a finite element analysis program. The number of finite element analyses required is fairly small. This sequential linearization method is particularly suitable for solving mixed-discrete nonlinear programming problems arising in structural optimization.