Optimization, all parts required

Problem 2 (5 parts, 12 points): Given a point 20 ER, the shrink function is defined as the unique solution to the following optimization problem: shrink(ro) = arg min (:11 +5 (3 1o)?. *ER 2.1 (1 point): Explain why shrink(0) = 0. 2.2 (3 points): Suppose that y = (v1.1) is the unique solution to the con- strained optimization problem min 12+ (1 - 1) such that y21, 122-1- Explain why wi = shrink(xo). 2.3 (2 points): Write the Lagrangian C(y,x) corresponding to the constrained optimization problem min 9 +5 (31 1o)? such that y 2 11, 122-41- 1 Problem 2 continued: 2.4 (4 points): Assuming 1, > 0, find all points (y) satisfying the KKT conditions for the problem min y2 + (1 1o) such that 922 01, 922-21- HINT: It is helpful to split into the 3 cases {x1 > 0,42 >0},{11 > 0.42 =0}, and {x1 = 0,42 0}. Problem 2 (5 parts, 12 points): Given a point 20 ER, the shrink function is defined as the unique solution to the following optimization problem: shrink(ro) = arg min (:11 +5 (3 1o)?. *ER 2.1 (1 point): Explain why shrink(0) = 0. 2.2 (3 points): Suppose that y = (v1.1) is the unique solution to the con- strained optimization problem min 12+ (1 - 1) such that y21, 122-1- Explain why wi = shrink(xo). 2.3 (2 points): Write the Lagrangian C(y,x) corresponding to the constrained optimization problem min 9 +5 (31 1o)? such that y 2 11, 122-41- 1 Problem 2 continued: 2.4 (4 points): Assuming 1, > 0, find all points (y) satisfying the KKT conditions for the problem min y2 + (1 1o) such that 922 01, 922-21- HINT: It is helpful to split into the 3 cases {x1 > 0,42 >0},{11 > 0.42 =0}, and {x1 = 0,42 0}