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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