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Lect04

# Lect04 - EE553 LECTURE 4 LECTURE OUTLINE Principal Gradient...

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EE553 LECTURE 4 LECTURE OUTLINE Principal Gradient Methods Gradient Methods - Choices of Direction Gradient Methods - Choice of Stepsize Gradient Methods - Convergence Issues

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CONVERGENCE RESULTS CONSTANT STEPSIZE Let { x k } be a sequence generated by a gradient method x k +1 = x k + α k d k , where { d k } is gradient related. Assume that for some constant L > 0 , we have f ( x ) - ∇ f ( y ) L x - y , x, y n , Also, assume that there exists a scalar such that for all k 0 < α k (2 - ) |∇ f ( x k ) d k | L d k 2 . Then either f ( x k ) → -∞ or else { f ( x k ) } converges and every limit point of { x k } is stationary.
MAIN PROOF IDEA 0 α α∇ f(x k )'d k + (1/2) α 2 L||d k || 2 × α∇ f(x k )'d k α = | f(x k )'d k | L||d k || |2 f(x k + α d k ) - f(x k ) The idea of the convergence proof for a constant stepsize. Given x k and the descent direction d k , the cost difference f ( x k + αd k ) - f ( x k ) is ma- jorized by α f ( x k ) d k + 1 2 α 2 L d k 2 (based on the Lipschitz assumption; see descent lemma). Mini- mization of this function over α yields the stepsize α = |∇ f ( x k ) d k | L d k 2 This stepsize reduces the cost function f as well.

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PRINCIPAL GRADIENT METHODS x k +1 = x k + α k d k , k = 0 , 1 , . . . where, if f ( x k ) = 0 , the direction d k satisfies f ( x k ) d k < 0 , and α k is a positive stepsize. Principal example: x k +1 = x k - α k D k f ( x k ) , where D k is a positive definite symmetric matrix
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