# Optimization Methods

#### Summary

• Overview
• Aerodynamic Optimization
• Compexity of Real-Life Problems
• Splitting up the Problem
• Nature of the Search Space
• State-of-the-Art Methods

In the last decade or so methods have been developed for treating intrinsically difficult minimization problems, including dis-joint and non convex search spaces. These methods, based on non-analytical approaches, have found their way in the design of aerodynamic components and systems.

When the solution is known to be in a neighborood of an optimum, gradient methods are widely used.

Several methods and computer codes are available, that come as black boxes to be used in different problems. This class of methods clearly fails when a radical new solution is sought. Hence the interest in this new class of methods.

### Aerodynamic Optimization

The idea behind a `design’ procedure is to locate a satisfactory solution in the design space. A procedure is said to be efficient if the number of solutions evaluated is small as compared to the size of the solution space. The same procedure is said to be robust if it finds the minimum under search spaces that feature a complex topology.

#### Complexity of Real-Life Problems

With the increasing complexity of engineering problems the job of optimization becomes more challenging. Rather that designing the best component possible, the quest is now for the best configuration that can operate efficiently over a wide range of operating conditions.

In aerospace and turbomachinery design it is required to produce the best configuration that satisfies at the same time aero- thermodynamic, aeroelastic, mechanic, manifacturing, time and budget requirements.

#### Splitting up the Problem

In general, each engineering problem can be described by different subsets of problems. Ideally, the subproblems are disjoint, e.g. they depend on free parameters that do not appear in the larger problem. When this not occur, a decision has to be made as to how to choose the parameters and their constraints. For the method to produce good results the choice of the free parameters and of the objective function is of the utmost importance.

#### Nature of the Search Space

When the search space is known to be non-convex, the optimum cannot be approached by traditional gradient-based methods. Instead, the search may start from different initial points, and proceed according to some heuristic procedures. Even then, no guarantee exists of obtaining the global optimum.

Complex solution topologies (disjoint and non-convex) emerge when a problem is defined by a very large number of parameters (typically, several dozens to several hundreds, sometimes even more), with constraints on at least some of them.

#### State-of-the-Art Methods

In the attempt to overcome prohibitive complexity of large-scale problems, some heuristic search mechanisms have been developed. These include:

Special pre-conditioning techniques may be used to smooth the objective function, so that the search of a global minimum becomes more tractable.

One approach, called global continuation method, is based on a smoothing technique, which transforms the objective function into a smoother function with fewer local minima.

There is a staggering number of publications concerned with numerical optimization methods. This is a field of research on its own.

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