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Let f : R" IR" be C and bounded below on IR", and Vf is globally Lipschitz on IR", i.e., there L > 0 such that IIV/(x) - Vf(y)||2 S Lilz - y||2 for any x.y. Consider a linear search method for the constrained optimization problem min f(x). (1) Let B € be an invertible matrix. Show that 1)B=ll2 > for any x € R". (2) Let (IX) be a sequence generated by the linear search method, where each descent direction dk = B, for a symmetric positive definite matrix Bk, and each step an satisfies the Wolfe conditions. Suppose there is exists a positive constant Y such that ||Bx||2||B;¹ ||2 for all k. Use Zoutendijk condition to show that limg /Vf(r))ll2 =0.

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