Download e-book for kindle: A cookbook of mathematics by Vinogradov V.

By Vinogradov V.

ISBN-10: 8086286207

ISBN-13: 9788086286204

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Example text

M ) = f (x1 , . . , xn ) + j−1 is the Lagrangian function of a non-linear program. The minimization version of Kuhn-Tucker necessary conditions is ∂L ∂L ≥ 0, xi ≥ 0 and xi = 0, ∂xi ∂xi i = 1, . . , n, 49 ∂L ∂L ≤ 0, λj ≥ 0 and λj = 0, ∂λj ∂λj j = 1, . . , m Note that, in general, the KT conditions are neither necessary nor sufficient for a local optimum. However, if certain assumptions are satisfied, the KT conditions become necessary and even sufficient. Example 70 The Lagrangian function of the nonlinear program in Example 68 is: L = x1 (10 − x1 ) + x2 (20 − x2 ) − λ1 (5x1 + 3x2 − 40) − λ2 (x1 − 5) − λ3 (x2 − 10).

Let define the Jacobian matrix of f 1 , . . , f m with respect to y1 , . . , ym as   J =   ∂f 1 ∂y1 ... ∂f 1 ∂ym ∂f m ∂y1 ... ∂f m ∂ym .. ..    .  Given the definition of Jacobian, we are in a position to formulate the general result: Proposition 25 (The General Implicit Function Theorem) Suppose f 1 , . . , f m are C (k) -functions in a set D ⊂Rn+m . ∗ Let (x∗ , y ∗ ) = (x∗1 , . . , x∗n , y1∗ , . . , ym ) be a solution to (3) in the interior of A. 36 Suppose also that det(J ) does not vanish at (x∗ , y ∗ ).

Example 40 For u ∈ V , let x = (x1 , . . , xn ) be the coordinate vector of u relative to B and y = (y1 , . . , yn ) is the coordinate vector of u relative to C. If D is the change-of-basis matrix relative to B, C then y = Cx. The change-of-basis matrix relative to C, B is D−1 . Example 41 Given the following bases for R2 : B = {(1, 1) , (1, 0) } and C = {(0, 1) , (1, 1) }, find the change-of-basis matrix D relative to B, C. The columns of D are the coordinate vectors of (1, 1) , (1, 0) relative to C.

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A cookbook of mathematics by Vinogradov V.


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