(Compressed Sensing) Consider the following linear regression problem with a weighted regularization term for a fixed scalar weight > O: n minimizex f(X) := |Ax b|> + i Z %17, j=1 where A is a m X n data matrix, b(# 0) is an m-dimension measuring data vector, and parameter 0 > m. If p = 1, the model is called LASSO (least absolute shrinkage and selection operator) that was originally introduced in geophysics and later by Tibshirani who coined the name. Note that the regularization function is differentiable everywhere except at x; = 0, and it becomes nonconvex when p # 0 for index ;. (a) What is the necessary condition on the first-order partial derivative z?Tf. |x* ? J (b) What is the necessary condition on the second-order partial derivative 32f| 0 xX* - 32xj (c) What is a second-order sufficient condition on V> f(x*)