Problem 4. (10pt) This problem helps us review some facts in the Gradient Descent method. Consider the Gradient Descent method for the optimization problem min f(x). XERO Suppose the initial guess is xo ER. Define the set S= {x ER" | f(x) = f(xo)}. Suppose also that f is strongly convex: all the eigenvalues of Vaf is bounded below uniformly by some m > 0. 1. Show that f(y) 2 f(x) + Of(x)"(y ) + " . ||1y = x || for all x, Y ES. 2. Let x* be the optimizer of the problem. Show that 1 f(x*) > f (x) - || f(x) ||3 for all x ES. 2m 3. Show that S is bounded. That is, there exists a constant C for which ||y||2 0. 1. Show that f(y) 2 f(x) + Of(x)"(y ) + " . ||1y = x || for all x, Y ES. 2. Let x* be the optimizer of the problem. Show that 1 f(x*) > f (x) - || f(x) ||3 for all x ES. 2m 3. Show that S is bounded. That is, there exists a constant C for which ||y||2