172B-3 - Readings Economics 172B Introduction to Operation...

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Economics 172B troduction to Operation Research Introduction to Operation Research 3. Unconstrained Optimization pring 010 Spring 2010 Herb Newhouse 1 eadings Readings illier & Lieberman (8 th dition) Hillier & Lieberman (8 edition) – 12.4: One-Variable Unconstrained ptimization Optimization. – 12.5: Multivariable Unconstrained ptimization. Optimization. 2 utline Outline omputational Methods for Unconstrained Computational Methods for Unconstrained Optimization: he Bisection Method – The Bisection Method. – Newton’s Method. he Zig ag Gradient Search – The Zig-Zag Gradient Search. 3 Unconstrained Optimization (Functions of One Variable)   max where is differentiable and concave. x fx First order condition: ' 0 x   e can use computational methods to appr ximate a  We can use computational methods to approximate a solution if ' 0 cannot be solved analytically. 4
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he Bisection Method The Bisection Method The bisection method works as long as the following hold:  '0 i f * i f * fx xx x     i f * f  These conditions hold for concave functions that are maximized at a finite value for *. x They also hold fo   r some other functions. The graphical example will have a small convex region. 5 he Basic Idea The Basic Idea Pick an on each side of the maximum 1. Pick an on each side of the maximum. . Cut the interval in half. x 2. Cut the interval in half. 3. Evaluate the sign of the derivative at the midpoint. 4. Create a new interval. 5. Repeat until you get a very small interval. 6 raphical Example Graphical Example * x 7 fficial Method (from H&L) Official Method (from H&L) Notation: ' current trial solution current lower bound on * current upper bound on * x x x x x pp error tolerance for * x 8
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    : Choose , and such that ' 0, ' 0. Initialization x x f x f x  The initial trial solution is ' .
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This note was uploaded on 06/08/2010 for the course ECON 171 taught by Professor Newhouse during the Spring '07 term at UCSD.

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172B-3 - Readings Economics 172B Introduction to Operation...

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