Chapter 10 - Numerical Methods for Unconstrained Optimum Design

Abstract

This chapter discusses the concepts and provides description of methods for unconstrained nonlinear optimization problems. The idea of iterative numerical algorithms is introduced to search for optimum solutions for the design problem. The algorithms are initiated with an estimate for the optimum solution that is improved iteratively if it does not satisfy the optimality conditions. There are two classes of methods for unconstrained and constrained problems that are gradient-based and direct search methods. Gradient-based search methods are iterative where the same calculations are repeated in every iteration. In such approaches, initial design is estimated and is improved until optimality conditions are satisfied. Many numerical methods have been developed for nonlinear programming problems. Gradient-based search method as the name implies, use gradients of the problem functions to perform the search for the optimum point. Therefore, all of the problem functions are assumed to be smooth and at least twice continuously differentiable everywhere in the feasible design space. In direct search methods only the function values are used in the search process.