Review - Review for Exam 2 1 Optimization conditions...

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1 Review for Exam 2
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2 Optimization conditions Definitions of Global and local minima We want to find a global but only afford to have local Unconstrained optimization problem KT condition (f’ = 0) 2 nd order necessary condition (f’’ PSD) Sufficient condition (f’’ PD) Condition for global minimum Convex objective on convex constraint set When the obj and constraint set become convex? Equality constrained problem Introduce Lagrangian KT condition   0, 0 LL x   ( , ) ( ) ( ) ii L f h x x x
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3 Optimization conditions Inequality constrained problem Introduce slack variables: KT condition Complementary slackness ( i g = 0) 2 nd -order necessary condition is P.S.D. for all feasible directions Sufficient condition is P.D. for all feasible directions  2 ( , , ) ( ) ( ) L s f xx 0, 0 x 2 x 2
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4 Numerical Method for Optimization Basic algorithm Move from one design to another until can’t reduce objective further Need function values (objective & constraints) and their gradient Need to find search direction and step size Unconstrained problem Descent condition: New objective function must be smaller than previous one Line search: find a k that minimize the objective function for given direction Step size termination criterion:  ( ) ( ) kk xd  ( ) ( ) 0 cd  a  () minimize ( ) ( ) f  ( 1) ( ) 0
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5 Numerical Method for Optimization Search direction Search direction should reduce the objective function Different algorithms are available for different ways of
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This note was uploaded on 02/15/2012 for the course EAS 6939 taught by Professor Staff during the Spring '08 term at University of Florida.

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Review - Review for Exam 2 1 Optimization conditions...

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