Engineering Optimization

9Second‐Order Model‐Based Optimizers: SQ and NR

9.1 Introduction

This chapter presents two search algorithms that use second‐order models of the surface: successive quadratic (SQ) and Newton–Raphson (NR). They presume more about the surface than the gradient‐based optimizers. Consequently these are faster when the surface is compatible with the algorithm concepts, which include continuum deterministic surfaces, no flat spots, and an initial trial solution in the vicinity of the optimum. These optimization approaches often are accepted as the premier optimization methods, and they are components in next‐level gradient‐based optimizers. So, they need to be presented, even though I find that they are wholly inappropriate for many applications, which have features that are inconsistent with the concepts on which these algorithms are predicated.

Although the examples and analysis will primarily be in 2‐D applications, SQ and NR are applicable to N‐D situations. The objective is

(9.1)

9.2 Successive Quadratic

This is a direct extension of the SQ presentation in Chapter 4 and is often termed a surrogate model method. In this case the surrogate model of how the function depends on the DV values is a quadratic relation. The model is used to approximate the function, and the next estimate of DV* is based on the model, not the function. It is very easy to obtain DV* values from a quadratic ...