MATH 4420 Midterm: Math4420 Midterm
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G(x) = (bt + b)x and 2g(x) = bt + b. [you may consider an example rst, say b = (cid:20)1 2. 3 4(cid:21)]: consider the unconstrained minimization problem for the quadratic function f (x) = 2 xt ax bt x + d xk+1 = xk + kpk, pk = f (xk), k = arg min. >0 f (xk + pk) (a) show that. F (xk)pk (pk)t apk (b) show that xk+1 xk is orthogonal to xk xk 1: (a) sketch contour lines for the function and determine x. Which minimizes f . f (x1, x2) = x2. 2 4x1 8x2 (b) apply the bfgs algorithm (page 140 in nocedal & wright) choosing x line x = x. 0 = (0, 0)t and h 0 = i, and show that the (c) show that if the method of steepest descent is started at x of steps. 0 = (0, 0)t it cannot converge to x.
