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Unformatted text preview: (9.18) − k ∇ Φ ( z ) k ≤ [ ∇ Φ ( z )] T ∆ z ≤ k ∇ Φ ( z ) k and minimum value [ ∇ Φ ( z )] T ∆ z can attain when ∆ z is restricted within the unit ball equals − k ∇ Φ ( z ) k . In fact, the equality will hold if and only if ∆ z is colinear with ∇ Φ ( z ) , i.e. (9.19) ∆ b z = − ∇ Φ ( z ) k ∇ Φ ( z ) k Thus, unit direction ∆ b z given by the above equation is the direction of steepest or maximum descent in which Φ ( z + ∆ z ) − Φ ( z ) reduces at the maximum rate. Algorithm Given a point z ( z ) , the steepest descent method performs a line search in the direction of − ∇ Φ ( z ) k ∇ Φ ( z ) k , i.e. the direction of steepest descent. INITIALIZE: z (0) , ε, k max , λ (0) k = 0 δ = 100 ∗ ε WHILE [( δ > ε ) AND ( k < k max )] s ( k ) = ∇ Φ ( z ( k ) ) k ∇ Φ ( z ( k ) ) k...
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.
 Fall '11
 Mr.Shaw
 Chemical Engineering, Optimization

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