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 --

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