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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.

** Next:** Review of Fourier Analysis
** Up:** Theoretical Tools for Problem
** Previous:** Theoretical Tools for Problem
Shlomo Ta'asan
2001-08-22