615.14 - Introduction to Numerical Optimization...

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Introduction to Numerical Optimization Biostatistics 615/815 ecture 14 Lecture 14
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Course is More Than Half Done! z If you have comments… z … they are very welcome Lectures Lecture notes Weekly Homework Midterm Content
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Last Lecture z Computer generated “random” numbers z Linear congruential generators provements through shuffling summing Improvements through shuffling, summing portance of using validated generators z Importance of using validated generators Beware of problems with the default rand() f nction uco
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Today … z Root finding z Minimization for functions of one variable z Ideas: Limits on accuracy Local approximations
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Numerical Optimization z Consider some function f(x) e.g. Likelihood for some model … z Find the value of x for which f takes a aximum or minimum value maximum or minimum value aximization and minimization are equivalent z Maximization and minimization are equivalent Replace f(x) with –f(x)
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Algorithmic Objectives z Solve problem… Conserve CPU time Conserve memory z Most often, the CPU time is dominated by the cost of evaluating f(x) Minimize the number of evaluations
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The Minimization Problem
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Specific Objectives z Finding global minimum The lowest possible value of the function Extremely hard problem z Finding local minimum Smallest value within finite neighborhood
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Typical Quality Checks z When solving an optimization problem it is good practice to check the quality of the solution z Try different starting values … z Perturb solution and repeat …
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A Quick Detour z Consider the problem of finding zeros for f(x) z Assume that you know: Point a where f(a) is positive Point b where f(b) is negative f(x) is continuous between a and b z How would you proceed to find x such that f(x)=0?
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double zero (double (* func )(double), double lo , double hi , double e ) { while (true) { ouble i o double d = hi lo ; double point = lo + d * 0.5; double fpoint = (* func )( point ); if ( fpoint 0.0) ( po t .) { d = lo point ; lo = point ; } else { d = point hi
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This note was uploaded on 12/26/2011 for the course BIO 615 taught by Professor Abecasis during the Fall '10 term at University of Michigan.

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615.14 - Introduction to Numerical Optimization...

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