Introduction

In this lecture we demonstrate the use of Fourier analysis to obtain a practical quantitative information about different optimization problems. Our main concern here is to study question regarding the formulation of the optimization problem, the choice of design variables and the choice of the cost functional. The complexity of the optimization problem will be shown to depend on these choices. We give examples to demonstrate this point and suggest a general approach for choosing design variables, and/or cost functional to achieve minimization problems that are good on one hand but also easy to solve on the other hand. For many engineering problems there is some freedom in the formulation of the problem and it is certainly an advantage to deal with the easier problems yet keeping the same engineering design tasks. We review basic facts from Fourier analysis and pseudo-differential operators and show its practical use for the analysis of Hessians. This analysis gives a very simple classification of problems based on the asymptotic behavior of the symbol of the Hessian at the high frequency range. It distinguishes between ill-posed (bad) problems, well-posed (good) problems, easy problems and difficult problems. This classification is of practical importance in the problem setup.