1. Solve Problem 7.7 in the Convex Optimization book (Boyd-Vandenberghe). [You don’t need to prove the hints] 2. Solve Problem 8.24 in the Convex Optimization book (Boyd-Vandenberghe). 3. Solve Problem 8.25 in the Convex Optimization book (Boyd-Vandenberghe). 4. Consider a differentiable function f : R n → R that is m-strongly convex. Show that for any x and y in R n we have (∇f(x) − ∇f(y))(x − y) ≥ mkx − yk 2 5. Consider a function f : R n → R that is M-smooth. (a) Show that for any x and y in R n and α ∈ [0, 1] we have f(αx + (1 − α)y) ≥ αf(x) + (1 − α)f(y) − α(1 − α)M 2 kx − yk 2 (b) Show that if the function f is differentiable then for any x and y in R n we have (∇f(x) − ∇f(y))(x − y) ≤ Mkx − yk 2 1
