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7

OPTIMIZATION

Optimization involves finding the minimum/maximum of an objective function
f(x) subject to some constraint x ∈ S. If there is no constraint for x to sat-
isfy—or, equivalently, S is the universe—then it is called an unconstrained
optimization; otherwise, it is a constrained optimization. In this chapter, we
will cover several unconstrained optimization techniques such as the golden
search method, the quadratic approximation method, the Nelder–Mead method,
the steepest descent method, the Newton method, the simulated-annealing (SA)
method, and the genetic algorithm (GA). As for constrained optimization, we
will only introduce the MATLAB built-in routines together with the routines for
unconstrained optimization. Note that we don’t have to distinguish maximization
and minimization because maximizing f(x) is equivalent to minimizing −f(x)
and so, without loss of generality, we deal only with the minimization problems.

7.1 UNCONSTRAINED OPTIMIZATION [L-2, CHAPTER 7]

7.1.1 Golden Search Method
This method is applicable to an unconstrained minimization problem such that
the solution interval [a, b] is known and the objective function f(x) is unimodal
within the interval; that is, the sign of its derivative f (x) changes at most once in

o
o
[a, b]sothat f(x) decreases/increases monotonically for [a, x ]/[x ,b], where
o
x is the solution that we are looking for. The so-called golden search procedure is
summarized below and is cast into the routine “opt_gs()”. We madeaMATLAB
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Applied Numerical Methods Using MATLAB , by Yang, Cao, Chung, and Morris
Copyright  2005 John Wiley & Sons, I nc., ISBN 0-471-69833-4

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