By Eugene Lerman
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Extra info for Multilinear Algebra notes [Lecture notes]
5) where L is the Lipschitz constant of the multidimensional function F(y). 2) by using algorithms proposed for minimizing functions in one dimension (see [74, 77, 100, 103, 132, 134, 136, 139, 140]). If such a method uses an approximation of the Peano curve pM (·) of level M and provides a lower bound UM∗ for the one-dimensional function y(x), then this value will be a lower bound for the function F(y) but only along the curve pM (·). Naturally, the following question becomes very important in this connection: Can we establish a lower bound for the function F(y) over the entire multidimensional search region D?
2), was defined by establishing a correspondence between the subintervals d(z1 , . . 6) and the subcubes D(z1 , . . , zM ) of each Mth partition (M = 1, 2, . ) and assuming that the inclusion x ∈ d(z1 , . . , zM ) induces the inclusion y(x) ∈ D(z1 , . . , zM ). Therefore, for any preset accuracy ε , 0 < ε < 1, it is possible to select a large integer M > 1 such that the deviation of any point y(x), x ∈ d(z1 , . . , zM ), from the center y(z1 , . . , zM ) of the hypercube D(z1 , . . 37) will not exceed ε (in each coordinate) because |y j (x) − y j (z1 , .
ZM ) of this coordinate. Then for any k, 1 ≤ k ≤ N, lk (x) = yk (z1 , . . 37). Now, it is left to outline the scheme for computing the number ν . Represent the sequence z1 , . . , zM as z1 , . . , zμ , zμ +1 , . . , zM where 1 ≤ μ ≤ M and zμ = 2N − 1, zμ +1 = . . = zM = 2N − 1; note that the case z1 = . . = zM = 2N − 1 is impossible because the center y(2N − 1, . . , 2N − 1) does not coincide with the node yq , q = 2MN − 1. As it follows from the construction of y(x), the centers y(z1 , .