Read e-book online Algorithms for Constrained Minimization of Smooth Nonlinear PDF

By A.G. Buckley, J-.L. Goffin

ISBN-10: 3642008127

ISBN-13: 9783642008122

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Extra resources for Algorithms for Constrained Minimization of Smooth Nonlinear Functions

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Z) where the old notation is retained for the new vectors. 36) or as Z = f ViXu Λ = ZF,. 36') G/ve/i a system ofn independent vectors Xl9 X2, . . , Χή in En, we can form a system of vectors Yl9 Y2, . . , Yn, such that the two systems {Xj} and {FJ are biorthogonal If * i = (*il> xi2> · · · > *in) a n d 7 i = CVil* 7i2» · · · » Jin), formation of the system {7J from the system {XJ amounts to formation of the inverse \\y{j \\}^ of the matrix libili;] 1 . (See Chapter IV regarding orthogonal and biorthogonal systems of functions).

THEOREM 5. f(x) on X as n -+- 00 are continuous, the limit function f(x) is also continuous on X. It follows from this that, if the limit function is discontinuous, the convergence as n -. 00 of the sequence {fn(x)}. is non-uniform. n EXAMPLE 22. The sequence {fn(x)} = {x } is uniformly convergent as n -+- 00 to f(x) = 0 on any segment [0, q], 0 <: q <:: 1; on the segment [0, 1] this sequence converges non-uniformly to the function 0 f(x) = { 1 for 0 for x ~ x = 1. )(- O-----a-------~b FIG.!. CAUCHY'S TEST.

Xk» of degree k « kx + kt+ . . ±kn. e. the number of dimensions of the system) is (zi+fc-1)! 2! - 10 (the single terms are x\, x\xZi x\xZi χγχ\, ΧχΧ^χ^ xxx\, xj, x\xZi x^c\y χ%). If a system Ln has one basis (el9 e2, . . , e^), it has an infinite set of bases. 9 e^) be another basis; then the elements ei of the old basis are expressible in terms of the elements e\ of the new in accordance with n *i = Σ αΰ*ί (/ = 1, 2, . . , TI), where the determinant | «gli;* ^ 0. An element/is expressible in terms of the elements of the new basis in accordance with / = Σ Ae\.

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Algorithms for Constrained Minimization of Smooth Nonlinear Functions by A.G. Buckley, J-.L. Goffin

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