Chemical Engineering Hand Written_Notes_Part_104

Chemical Engineering Hand Written_Notes_Part_104 - (9.18) k...

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210 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Figure 7. Level surfaces for n = 2 Φ ( z )= Φ ( z + z ) ' Φ ( z )+[ Φ ( z )] T z = C (9.12) z =( z z ) (9.13) This is equation of the plane tangent to surface Φ ( z ) at point z . The equation of level surface through z is (9.14) C = Φ ( z )= Φ ( z ) Combining above two equations are obtain the equation of tangent surface at z = z as (9.15) ( z z ) T Φ ( z )=0 Thus, gradient at z = z is perpendicular to the level surface passing through Φ ( z ) (See Figure 8). We will now show that it points in the direction in which the function in- creases most rapidly, and in fact, Φ ( z ) is the direction of maximum slope. If [ Φ ( z )] T z < 0 then (9.16) Φ ( z + z ) < Φ ( z ) and z is called as descent direction. Suppose we f x our selves to unit sphere in the neighborhood of z = z i.e. set of all z such that k z k 1 and want to
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9. UNCONSTRAINED NONLINEAR PROGRAMMING 211 Figure 8 f nd direction z such that ∆Φ ( z ) T z algebraically minimum. Using Cauchy- Schwartz inequality together with k z k 1 , we have (9.17) ¯ ¯ ¯ [ Φ ( z )] T z ¯ ¯ ¯ k Φ ( z ) kk z k k Φ ( z ) k This implies
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Unformatted text preview: (9.18) k ( z ) k [ ( z )] T z k ( z ) k and minimum value [ ( z )] T z can attain when z is restricted within the unit ball equals k ( z ) k . In fact, the equality will hold if and only if z is colinear with ( z ) , i.e. (9.19) b z = ( z ) k ( z ) k Thus, unit direction b z given by the above equation is the direction of steepest or maximum descent in which ( z + z ) ( z ) reduces at the maximum rate. Algorithm Given a point z ( z ) , the steepest descent method performs a line search in the direction of ( z ) k ( z ) k , i.e. the direction of steepest descent. INITIALIZE: z (0) , , k max , (0) k = 0 = 100 WHILE [( &gt; ) AND ( k &lt; k max )] s ( k ) = ( z ( k ) ) k ( z ( k ) ) k...
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Chemical Engineering Hand Written_Notes_Part_104 - (9.18) k...

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