Given f:^n rightarrow, f^1 consider the algorithm x^(k + 1) = x^(k) + alpha_k d(

Question

Given f:^n rightarrow, f^1 consider the algorithm x^(k + 1) = x^(k) + alpha_k d(k) where d^(1), d^(2), .. ., are vectors in^n and alpha_k Greaterthanorequalto 0 is chosen to minimize f(x^(k) + alpha d^(k)) that is, alpha_k = argmin alpha Greaterthanorequalto 0 f (x^(k) + alpha d^(k)). Note that the general algorithm above encompasses almost all algorithms that we discussed in this part, including the steepest descent, Newton, conjugate gradient, and quasi-Newton algorithms. Let g^(k) = f(x)^(k)), and assume that d^(k)T g^(k) 0. Show that d^(k)T g^(k+1) = 0 d. Show that the following algorithms all satisfy the condition d^(k)T g^(k) 0 For the case where f(x) = 1/2 x^TQx - x^T b, with Q = Q^T > 0, derive an expression for alpha_k in terms of Q, d^(k), and g^(k).

Explanation / Answer

features to be added;

problems to be corrected;

classes to implement;

place markers for error-handling code; and

reminders to check in the file.

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