fundamental-engineering-optimization-methods.pdf

# Estimating the minimum once the minimum has been

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Estimating the Minimum. Once the minimum has been bracketed to a small interval, a quadratic or cubic polynomial approximation is used to find the minimizer. If the polynomial minimizer ߙ satisfies Wolfe’s condition for the desired ߟ value ´VD\ ߟ ൌ ͲǤͷሻ and the sufficient decrease condition for the desired ߤ value (say ߤ ൌ ͲǤʹ ), it is taken as the function minimizer, otherwise ߙ is used to replace one of the ߙ ߙ ³ and the polynomial approximation step repeated. Quadratic curve Fitting. Assuming that the interval ሾߙ ǡ ߙ contains the minimum of a unimodal function, ݂ሺߙሻ ³ it can be approximated by a quadratic function: ݍሺߙሻ ൌ ܽ ൅ ܽ ߙ ൅ ܽ ߙ ² A quadratic approximation uses three points ሼߙ ǡ ߙ ǡ ߙ ³ where the mid-point of the interval may be used for ߙ ² The quadratic coefficients ሼܽ ǡ ܽ ǡ ܽ are solved from: ݂ሺߙ ሻ ൌ ܽ ൅ ܽ ߙ ൅ ܽ ߙ ǡ ߙ ߳ሼߙ ǡ ߙ ǡ ߙ ³ which results in the following expressions: ܽ ͳ ߙ െ ߙ ݂ሺߙ ሻ െ ݂ሺߙ ߙ െ ߙ ݂ሺߙ ሻ െ ݂ሺߙ ߙ െ ߙ ቉ Ǣ ܽ ͳ ߙ െ ߙ ൫݂ሺߙ ሻ െ ݂ሺߙ ሻ൯ െ ܽ ሺߙ ൅ ߙ ሻǢ (7.10) ܽ ൌ ݂ሺߙ ሻ െ ܽ ߙ െ ܽ ߙ The minimum for ݍሺߙሻ can be computed by setting ݍ ሺߙሻ ൌ Ͳǡ and is given as: ߙ ௠௜௡ ൌ െ ଶ௔ ² An explicit formula for ߙ ௠௜௡ in terms of the three interval points can also be derived and is given as: ߙ ௠௜௡ ൌ ߙ ͳ ʹ ሺߙ െ ߙ ሺ݂ሺߙ ሻ െ ݂ሺߙ ሻሻ െ ሺߙ െ ߙ ሺ݂ሺߙ ሻ െ ݂ሺߙ ሻሻ ሺߙ െ ߙ ሻሺ݂ሺߙ ሻ െ ݂ሺߙ ሻሻ െ ሺߙ െ ߙ ሻሺ݂ሺߙ ሻ െ ݂ሺߙ ሻሻ (7.11) An example of the approximate search algorithm is now presented.

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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 134 ±umerical Optimization Methods Example 7.1: Approximate search algorithm (Ganguli, p. 121) We wish to approximately solve the following minimization problem: ݂ሺߙሻ ൌ ݁ ିఈ ൅ ߙ ² We use Arjimo’s rule with: ߤ ൌ ͲǤʹ ³ and ߙ ൌ ͲǤͳǡ ͲǤʹǡ ǥ ³ to estimate the minimum. The Matlab commands used for this purpose and the corresponding results appear below: >> f=inline(‘x.*x+exp(-x)’); mu=0.2; al=0:.1:1; >> feval(f,al) 1.0000 0.9148 0.8587 0.8308 0.8303 0.8565 0.9088 0.9866 1.0893 1.2166 1.3679 >> 1-mu*al 1.0000 0.9800 0.9600 0.9400 0.9200 0.9000 0.8800 0.8600 0.8400 0.8200 0.8000 Then, according to Arjimo’s condition, an estimate of the minimum is given as: ߙ ൌ ͲǤͷǤ Further, since ݂ ሺͲሻ ൏ Ͳ DQG ݂ ሺߙሻ ൐ Ͳ ³ W and the minimum is bracketed by [0, 0.5]. We next use quadratic approximation of the function over ሼͲǡ ǡ ߙሽ to estimate the minimum as follows: al=0; ai=0.25; au=0.5;
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