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︰電機系
考試日期（年月日）︰109.04.23
考試時限（分鐘）：100
試題 :
（註：以下部分數學式以latex語法表示）
1. (32%) For each of the following functions, prove or disprove if it is a con-
vex function.
(a) (10%) f : R^2 -> R, dom f = R_{++}^2, f(x_1, x_2) = 1/(x_1 + 1/x_2).
(b) (12%) Let C \subseteq R^n be a subset of R^n. Define f_C : R^n -> R with
dom f_C = R^n as f_C(x) = sup _{y \in C} ||x-y||_2 and g_C : R^n -> R with dom
g_C = R^n as g_C(x) = inf_{y \in C} ||x-y||_2.
(i) (6%) If C is convex, prove or disprove if f_C and g_C are convex, respec-
tively (3% each).
(ii) (6%) If C is not convex, prove or disprove if f_C and g_C are convex, r-
espectively (3% each).
(c) (10%) h : R^n -> R, h(x) = (f(x))^2/g(x), with dom h = dom f \cap dom g
where f : R^n -> R and g : R^n -> R are both positive and convex within their
domains.
2. (10%) Suppose C \subseteq R^n is nonempty and not convex. Prove or disprove
that S = {y \in R^n | y^{T}x≦1 for all x \in C} is convex.
3. (8%) Let f(x) = logx with dom f = R_{++}. Find f^*, the conjugate function
of f, along with its domain, dom f^*.
4. (20%) Determine whether each of the following sets is a convex set. You don-
't have to write down the proofs. The score you get in this section is
s = max{0.5n_c - 10n_w} where n_c and n_w are the numbers of correct answers
and wrong answers (not including those left blank).
(a) (5%) An ellipsoid, defined as {x | (x_x_c)^{T}P^{-1}(x-x_c)≦1} for any gi-
ven x_c \in R^n and P \in S_{++}^n.
(b) (5%) {a \in R^k | p(0) = 1; |p(t)|≧1 for α≦t≦β}, where p(t) = a_1 +
a_{2}t + ... + a_{k}^{k+1}.
(c) (5%) {a \in R^k} | p(0) = 1; |p(t)|≦1 for t≦α or t≧β}, where p(t) =
a_1 + a_{2}t + ... + a_{k}t^{k-1}, and α<β.
(d) (5%) {x \in R^n | ||Ax+b||_2 ≦ c^{T}x + d}.
5. (30%) For the following optimization problems, determine whether each of th-
em is (1) a convex optimization problem, (2) an LP, (3) a QP, (4) a QCQP, (5) a
SOCP, (6) a quasi-convex optimization problem. Writh your answer as a table of
5 rows and 6 columns, with each entry being T (yes), F (no), or left blank. The
score you get in this section is s = max{0, n_c - 2n_w} where n_c and n_w are
the numbers of correct answers and wrong answers (not including those left bla-
nk).
(a) minimize c^{T}x
subject to x^{T}Px≦1
where P \in S_{++}^n.
(b) minimize x_1
subject to \sqrt{x_{1}^2 + 4x_{2}^2 + 9x_{3}^2}
≦ 2x_1 + x_2
(c) minimize (x_{1}^5 + x_{2}^5)^{1/5}
subject to x_1 + x_2 = 1
x_1 - x_2 ≦ 0
(d) minimize x_{1}^2 2x_{2}^2 + 3x_{3}^2
subject to -x_1 - x_2 - x_3 ≦ 1
(e) minimize (\Pi_{k=1}^{n} x_k)^{1/n}
subject to c^{T}x≧1
where c \in R^n