Homework 2 assignment for course ISYE6669 PDF

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ISyE 6669 HW 2

1. Expand the following summations: (For example, the answer to part (a) is x1 + x2 + x3 .) P3 P4 P3 (xi + yij ) (d) (a) i=1 xi P3i=1 j=2 P3 t (2k + 1)x 2 w (e) (b) P5k=1 Pn+3 k+1 P3t=1 Pi 2t x (f) (c) m=n+1 xn ym j=1 ij n=3 i=1     3 2 2. Consider the following two vectors: x =  1  and y =  1 . 2 4 (a) What is dimension of x and y? (b) Compute x + 2y. (c) Compute ||x − y||. (d) Compute x⊤ (x + y). (e) Compute x2 y1 . 3. Verify whether each of the following sets is convex or not: (a) X = {(x1 , x2 ) | x12 + x22 ≥ 0}. (b) X = {x | 3 − x2 = 0}. (c) X = {(x1 , x2 ) |

x1 (x2 +2)

≤ 3, x2 ≥ −1}.

4. Verify whether the following problems are convex programs or not (a) min x21 + x22 s.t. (x1 , x2 ) ∈ R2 . (b) max 3x1 + 2x2 s.t. x12 + x22 ≤ 10. Pn Pn i 2i s.t. (c) min i=1 2 (xi ) i=1 xi ≥ 10. 5. A quantity y is known to depend upon another quantity x. A set of n n has been collected. data pairs {yi , xi }i=1 (a) Formulate an optimization model for fitting the “best” straight line y = a + bx to the data set, where best is with respect to the sum of absolute deviations. What kind of an optimization model is it? (b) Re-formulate the optimization model in part (a) where best is with respect to the maximum absolute deviation. What kind of an optimization model is it? 1

(c) Formulate an optimization model for fitting the “best” quadratic curve y = a + bx + cx2 to the data set, where best is with respect to the maximum absolute deviations. What kind of an optimization model is it ?

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