IMSE 505 Homework DUE TIME: 11:59 p.m. Nov. 6 Your answers must be typed and clear in Word or Latex! Each problem answer must appear on separate sheets of paper. 1.(25 points) Minimize 1 2 Find a point satisfying the first-order necessary condition for a solution. Show that this point is a global minimum of f (x). Starting at ( , = (0, 0) and using the pure Newton method, what is the next point (,)? 2.(25 points) Minimize f (x) over Rn. Let xk and xk+1 be two con- secutive points generated by the steepest descent algorithm using the minimization rule. Show that f (xk) and f (xk+1) are orthogonal, i.e., f (xk)T f (xk+1) = 0. 3.(25 points) The function f (x) is differentiable and convex on Rn. Show that for any x, y Rn, (f (x) f (y))T (x y) 0 4.(25 points) Consider the regression model y = + x + rx2 + s, where x is the independent variable, y is the observed dependent variable, , , and are unknown parameters, and s is random component representing the experiment error. The following table gives the values of x and the corresponding values of y. x 0.6 1.3 2.2 3.1 3.9 4.5 5.0 5.2 y 3 3 -10 -25 -50 -80 -100 -98 Formulate the problem of finding the best estimates of , , and as an unconstrained optimization problem by minimizing the sum of squared errors. Code the Gauss-Newton method to solve the formu- lated optimization problem. You need to submit the source code and report the entire sequence of solutions {(k, k, k)}k0 attained at all iterations.