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）︰110.04.22 考試時限（分鐘）：120 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) (8%) f : R^n -> R with f(x) = - \Pi_{i=1}^{n} x_{i}^{αi} and dom f = R_{++}^n where 1^{T}α = 1. (b) (8%) f : R^n -> R with f(y) = sup_{x \in C} a^{T}x where a \in R^n is a constant vector and C = {x \in R^n | ||x||_2 = 1} (c) (8%) f : R^n -> R with f(y) = inf_{x \in D} ||y-x||_p where a \in R^n is a constant vector and D = {x \in R^n | ||x||_2 ≦1} (d) (8%) f : R^n -> R with f(x) = ||Ax-b||_{2}^{2} / (1-x^{T}x) and dom f = {x \in R^n | ||x||≦1} for any A \in R^{m*n} and b \in R^m 2. (19%) Consider the complex least l_p-norm problem minimize ||x||_p subject to Ax = b where A \in C^{m*n}, b \in C^m and the variable is x \in C^n. Here ||‧|| deno- tes the l_p norm on C^n, define as ||x||_p = (\sum_{i=1}^{n} |x_i|_^p)^{1/p} for p≧1 and ||x||_{infty} = max_{i=1,...,n} |x_i|. We assume A is full rank, and m < n. (a) (9%) Reformulate the complex least l_2-norm problem as a least l_2-norm pr- oblem with real problem data and variable. Hint: Use z = (Re(x), Im(x)) \in R^{2n} as the variable. (b) (9%) Reformulate the complex least l_{\infty}-norm problem as an SOCP. 3. (20%) Consider a linear program minimize c^{T}x subject to Ax \preceq b where c \in R^n is uncertain and lies in an ellipsoid ε = {\bar{c} + Pu | ||u||≦1} with P \in R^{n*n}. We intend to study the robu- st linear program and minimize the largest possible value of c^{T}x among all c \in ε, that is, minimize sup_{c \in ε} c^{T}x subject to Ax \preceq b (1) (a) (10%) Determine if Problem (1) is a convex problem. Why? (b0 (10%) Try to rewrite Problem (1) as an equivalent problem in the form of e- ither a QCQP or an SOCP. Explain why the new problem you write is a QCQP (or an SOCP). IF you find that it is not possible, please also explain why it is so. 4. (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. Write 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)^2 subject to x^{T}Px≦1 where P \in S_{++}^n. (b) minimize 2x_1 + 3x_2 subject to \sqrt{x_{1}^2 + 4x_{2}^2 + 9x_{3}^3} ≧ 4x_1 + 3x_2 (c) minimize x_{1}^2 + x_{2}^2 subject to x_1 + x_2 = 1 x_1 - x_2 ≦ 0 (d) minimize x_{1}^2 + 2x_{2}^4 + 3x_{3}^6 subject to x_1 + x_2 - x_3 ≦ 1 (e) maximize log(\sum_{k=1}^{n} x_k) subject to c^{T}x ≧ 1 where c \in R^n. --

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