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CHAPTER 2: OPTIMIZATION METHODOLOGY
2.14.2
Convergence to an optimal point
•
First-order approximations.
Because of the absence of curvature, it is likely that perhaps 5 to 10 iterations may be required for
convergence. The first-order approximation method turns out to be robust thanks to the successive
approximation scheme which addresses possible oscillatory behavior. Linear approximations may be
rather inaccurate to study trade-off, i.e., in general they make poor global approximations, but this is not
necessarily true and must be assessed using the error parameters.
• Second-order approximations.
Due to the consideration of curvature, a successive quadratic response surface method is likely to be
more robust, but can be more expensive, depending on the number of design variables.
• Other approximations.
Neural networks (Section 2.10.1) and Kriging (Section 2.10.2) provide good approximations when many
design points are used. A suggested approach is to start the optimization procedure in the full design
space, with the number of points at least of the order of the minimum required for a quadratic
approximation. To converge to an optimum, use the iterative scheme with domain reduction as with any
other approximations, but choose to update the experimental design and response surfaces after each
iteration (this is the default method for neural nets and Kriging). The response surface will be built using
the total number of points. At the end of the iterative optimization procedure, the trade-off can be based
on neural networks or Kriging surfaces using results from all the points calculated. See also Sections
2.10, 4.5.1 and 9.1.2.
• Random search methods. These methods can be used in very much the same way as first order
approximations, but are bound to be more expensive (Section 2.13).
2.15 Applications of optimization
2.15.1
Multicriteria Design Optimization
A typical design formulation is somewhat distinct from the standard formulation for mathematical
optimization (Eq. 2.3). Most design problems present multiple objectives, design targets and design
constraints whereas the standard mathematical programming problem is defined in terms of a single
objective and multiple constraints. The standard formulation of Eq. (2.3) has been modified to represent the
more general approach as applied in LS-OPT.
Minimize the function
p[ f ( x )]
(2.61)
subject to the inequality constraint functions
L j ≤ g j ( x ) ≤ U j ; j = 1,2,K, m
The preference function p can be formulated to incorporate target values of objectives.
Two methods for achieving this are given:
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