Lec8 - Optimization One-dimensional and multidimensional...

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Optimization One-dimensional and multi- dimensional unconstrained optimization
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Lecture 8 2 -4 -2 2 4 -1 -0.5 0.5 1 Optimization z Optimization is about finding maximum or minimum values of a single-variable function f(x), or a multiple variable function f(x 1 , …, x N ) z Optimization can be Local: find a local minimum or maximum in some small domain of scalar x or vector {x} Global: find the largest or smallest value of the function throughout a large domain of x or {x} z Optimization can also be constrained or unconstrained The constraints can limit the domain or trajectory of the search, or require that linear or nonlinear combinations of the {x} variables satisfy inequalities In this course we will only discuss unconstrained optimization f(x) x maximum minimum root
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Lecture 8 3 Optimization: Connections to Root-finding and Linear/Nonlinear Systems z Optimization is related to other subjects that we have already discussed: Root-finding: Notice that finding a local minimum or maximum of a function f(x) is the same as solving for a root of the equation Linear systems: If [A] is symmetric and positive definite we can solve the linear system [A]{x}={b} by finding the minimum of the multivariable function Nonlinear systems that can be expressed as the minimum/maximum of a function f({x}) can be solved with optimization methods z Optimization techniques are generally more robust and efficient than finding roots or solving large nonlinear systems Use them whenever possible!
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Lecture 8 4 One-dimensional unconstrained optimization: Golden Search Just as in root-finding, there are two general classes of 1-D optimization methods: -- bracketing (closed) methods , e.g. bisection or golden search -- open methods, e.g. Newton’s method Without loss of generality, we will discuss methods for finding a local maximum -- methods for finding local minima are very similar The Golden search method is similar to bisection, but makes a different choice for narrowing the search. It relies on a special number, the “Golden Ratio”: This ratio is close to the height/width ratio in certain ancient Greek buildings, e.g. the Parthenon
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Lecture 8 5 Golden Search Method z Start with two x-values, x l and x u , which bracket the maximum z Select two new interior points x 1 and x 2 using the golden ratio: z Evaluate f(x 1 ) and f(x 2 ) z If f(x 1 ) > f(x 2 ), x 2 becomes next x l z If f(x 2 ) > f(x 1 ), x 1 becomes next x u f x l x 1 x u x 2
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6 Golden Search Method: Next iteration z In the second iteration, we take special advantage of the golden ratio z Select two new interior points x 1 and x 2 using the golden ratio: z But we only need one new function evaluation, f(x 1 ), since the new x 2 is the old x 1 ! z
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This note was uploaded on 12/29/2011 for the course CHE 132b taught by Professor Ceweb during the Fall '09 term at UCSB.

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Lec8 - Optimization One-dimensional and multidimensional...

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