INTRODUCTION TO OPTIMIZATION

4. (10 pis.) Let rx, Xx+1 E R" be two consecutive estimates generated by a quasi- Newton method for twice-continuously differentiable function f : R" -+ R. In the quasi-Newton, you have the following secant equation where yx = V/(2:+1) - VS(z), s = 14+1 - 2x, and the symmetric matrix HA41 approximates (V'S(Ix+1) ) . Find a (symmetric) rank-one update formula of the form HA41 = He + ouu], and Hk. where a E R, u E R". In particular you need to determine quu" in terms of sk, k, 5. (15 pts.) Let f(31, 12) = 21-12 + In(r )-ez, Apply a step of trust-region algorithm based on Newton's method with zo = (1,0), An = 1, 7) = 0.2, and 72 = 0.8