7. Given f:R R,feC consider the algorithm (+1)= x(k)+ad(k) where dd),... are vectors in R,...

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7. Given f:R" →R,feC consider the algorithm 最佳化演算法與應用 (+1)= x(k)+ad(k) where d¹d²),... are vectors in R", and a 20 is chosen to minimize f(x) +ad)); that is, a = arg min f(x) + ad()) a20 Page 2 of 3 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) = f(x)), and assume that dg(*) <0. Show that the following algorithms all satisfy the condition dkg) <0, ifg() #0: (1) Steepest descent algorithm. (2) Newton's method, assuming that the Hessian is positive definite. (3) Conjugate gradient algorithm. (4) Quasi-Newton algorithm, assuming that H, >0. 7. Given f:R" →R,feC consider the algorithm 最佳化演算法與應用 (+1)= x(k)+ad(k) where d¹d²),... are vectors in R", and a 20 is chosen to minimize f(x) +ad)); that is, a = arg min f(x) + ad()) a20 Page 2 of 3 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) = f(x)), and assume that dg(*) <0. Show that the following algorithms all satisfy the condition dkg) <0, ifg() #0: (1) Steepest descent algorithm. (2) Newton's method, assuming that the Hessian is positive definite. (3) Conjugate gradient algorithm. (4) Quasi-Newton algorithm, assuming that H, >0.

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1 Steepest descent algorithm The steepest descent algorithm is a iterative optimization algorithm th... View the full answer