Engineering Optimization

17Convergence Criteria 2: N‐D Applications

17.1 Introduction

This chapter builds on Chapter 6 and elaborates on convergence criteria that can be used in optimization. The concepts can be classified as either deterministic or stochastic and as either single player or multiplayer applications. Single trial solution searches (such as NR, LM, CHD, etc.) have one trial point and seek to move it. Here, convergence is based on the sequential trial solution, point‐to‐point changes. Multiplayer searches (such as LF and PSO) and pattern‐type optimizers (such as HJ and NM) have a cluster of trial solutions. Here, convergence would be based on the range of the cluster or pattern.

However, there is a lead player in the pattern‐type and multiplayer optimizers, and ignoring the other active players in a multiplayer procedure, the convergence criteria could be based on the iteration‐to‐iteration changes in the best in the cluster. So, the single TS convergence criteria could be applied to a multiplayer algorithm.

17.2 Defining an Iteration

Convergence is tested after each iteration. In many optimizer procedures the definition of what constitutes an iteration seems obvious. An iteration is when one complete cycle of analysis and consequential TS move is complete. In CHD an iteration constitutes N surface explorations, one for each dimension, each DV. In HJ, an iteration has an average of 1.5N + 1 surface explorations. In CSLS, there are N + 1 function evaluations to define the univariate search ...