課程名稱︰凸函數最佳化
課程性質︰電機所選修
課程教師︰蘇柏青
開課學院:電資學院
開課系所︰電機所
考試日期(年月日)︰108.06.20
考試時限(分鐘):100
試題 :
註:以下部分數學符號與式子以LaTeX語法表示。
1. (15%) Consider the convex unconstrained optimization problem whose variable
is x \in R^2:
minimize f_0(x) = [x_1 x_2][5 1 \\ 1 5][x_1 x_2].
We will study some types of descent methods in this problem.
(a) (3%) Find \nabla f_0, the gradient of f_0 for any x \in R^2.
(b) (4%) Find \nabla^2f_0, the Hessian of f_0 for any x \in R^2.
(c) (3%) Suppose the initial point is chosen to be x^{(0)} = [3 2]^T. Find the
gradient descent direction \Delta x_{gd}
(d) (5%) Again, let the initial point be x^{(0)} = [3 2]^T. Find the Newton st-
ep \Delta x_{nt}
2. (45%) Consider the convex piecewise-linear minimization problem
minimize \max_{i=1,...,m} (a_i^Tx + b_i)