Problem 1 Strong Convexity - (25 pts). Consider a twice differentiable function f(x) with r ER". The function f is said to be strongly convex if and only if there exists an m > 0 such that V's(x) - ml (1) is positive semidefinite (or written as V2f(x) > m/ by convention) for all r ( R", or equivalently, if and only if f() 2f(2)+V/(2) (y-2)+ ,ly-2; (2) for all y,x e R". Suppose that f (x) is strongly convex and there exists an optimal point a* := arg minzer. f(x) and f* = f(2*). 1. Show that f(x) - f* 7 (V/(a); for all x e R". (Hint: use the inequality (2).) 2. Show that la - r* |/2 )+ , lly -zz, for an M > m. Show that for the minimizing sequence {} generated by the gradient descent method with exact line search, there holds f(241)