L15_numex - Lecture 15: Numerical Experiments with...

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Lecture 15: Numerical Experiments with Algorithms for Unconstrained Optimization March 12, 2007
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Lecture 15 Outline Gradient Descent Method Sufficient Function Decrease Gradient and Newton’s Method (Homework) Condition Number Affine Transformation Convex Optimization 1
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Lecture 15 Gradient Descent Method Example : Consider minimizing following function over R f ( x ) = 3 4 (1 - | x | ) 2 - 2(1 - | x | ) | x | > 1 x 2 - 1 | x | ≤ 1 Gradient Descent Method with Succesive Reduction Stepsize: Given a starting point x dom f , a stepsize α > 0 , and a stepsize reduction parameter β (0 , 1) Repeat 1. Direction: compute the gradient f ( x ) 2. Succesive Stepsize Decrease: Set λ = α . Function Evaluation: Compute f ( x + λd ) . Function Decrease Test: If f ( x + λd ) f ( x ) set λ = βλ and go to Function Evaluation, otherwise go to Update 3. Update: compute new point x := x + αd Until A stopping criterion is satisfied Convex Optimization 2
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L15_numex - Lecture 15: Numerical Experiments with...

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