Final Exam Review

Final Exam Review - EAD 115 Numerical Solution of...

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EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science
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Multidimensional Unconstrained Optimization • Suppose we have a function f() of more than one variable f(x 1 , x 2 , …, x n ) • We want to find the values of x 1 , x 2 , …, x n that give f() the largest (or smallest) possible value • Graphical solution is not possible, but a graphical picture helps understanding • Hilltops and contour maps
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Methods of solution • Direct or non-gradient methods do not require derivatives – Grid search – Random search – One variable at a time – Line searches and Powell’s method – Simplex optimization
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• Gradient methods use first and possibly second derivatives – Gradient is the vector of first partials – Hessian is the matrix of second partials – Steepest ascent/descent – Conjugate gradient – Newton’s method – Quasi-Newton methods
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Grid and Random Search • Given a function and limits on each variable, generate a set of random points in the domain, and eventually choose the one with the largest function value • Alternatively, divide the interval on each variable into small segments and check the function for all possible combinations
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Features of Random and Grid Search • Slow and inefficient • Requires knowledge of domain • Works even for discontinuous functions • Poor in high dimension • Grid search can be used iteratively, with progressively narrowing domains
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Line searches • Given a starting point and a direction, search for the maximum, or for a good next point, in that direction. • Equivalent to one dimensional optimization, so can use Newton’s method or another method from previous chapter • Different methods use different directions
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12 (, , , ) , , ) () ( , , , ) ( ) n n n xx x vv v f fx x x g λf λ x v x xv 
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One-Variable-at-a Time Search • Given a function f() of n variables, search in the direction in which only variable 1, changes • Then search in the direction from that point in which only variable 2 changes, etc. • Slow and inefficient in general • Can speed up by searching in a direction after n changes ( pattern direction )
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Powell’s Method • If f() is quadratic, and if two points are found by line searches in the same direction from two different starting points, then the line joining the two ending points (a conjugate direction ) heads toward the optimum • Since many functions we encounter are approximately quadratic near the optimum, this can be effective
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• Start with a point x 0 and two random directions h 1 and h 2 • Search in the direction of h 1 from x 0 to find a new point x 1 • Search in the direction of h 2 from x 1 to find a new point x 2 . Let h 3 be the direction joining x 0 to x 2 • Search in the direction of h 3 from x 2 to find a new point x 3 • Search in the direction of h 2 from x 3 to find a new point x 4 • Search in the direction of h 3 from x 4 to find a new point x 5
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• Points x 3 and x 5 have been found by searching in the direction of h 3
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Final Exam Review - EAD 115 Numerical Solution of...

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