2. (60%) Consider the optimization problem minimize ZER fo(x) subject to ||Ax-b1 Se where A Rx,...

  

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2. (60%) Consider the optimization problem minimize ZER fo(x) subject to ||Ax-b1 Se where A Rx, b R", and e> 0, and fo(x) = length(r). (a) (10%) Determine if Problem (2) is a convex problem. Justify your answer. (2) (b) (10%) Show that Problem (2) is a quasiconvex problem. (Hint: You need to show that fo(r) is a quasiconvex function in x.) (c) (10%) Construct a family of functions o: R" R. such that and is convex for all t. (d) (10%) Let n 3 and m = 3. Let A = f(x) t(x) 0 123 4 5 6 7 8 9 b= --8 , and 0.1. Use CVX and the bisection method your learned in class to solve the problem. Write down the optimal value p* and an optimal point * you obtained from your algorithm. Make sure that the reported optimal point r satisfies the constraint inequality and yields the optimal value. 1 2 3 4 5678 (e) (10%) Repeat (2d) for n = 4, m = 5, A = 9 10 11 12 13 14 16 15 17 18 20 19 b= 1243 5 and e=0.1. 2. (60%) Consider the optimization problem minimize ZER fo(x) subject to ||Ax-b1 Se where A Rx, b R", and e> 0, and fo(x) = length(r). (a) (10%) Determine if Problem (2) is a convex problem. Justify your answer. (2) (b) (10%) Show that Problem (2) is a quasiconvex problem. (Hint: You need to show that fo(r) is a quasiconvex function in x.) (c) (10%) Construct a family of functions o: R" R. such that and is convex for all t. (d) (10%) Let n 3 and m = 3. Let A = f(x) t(x) 0 123 4 5 6 7 8 9 b= --8 , and 0.1. Use CVX and the bisection method your learned in class to solve the problem. Write down the optimal value p* and an optimal point * you obtained from your algorithm. Make sure that the reported optimal point r satisfies the constraint inequality and yields the optimal value. 1 2 3 4 5678 (e) (10%) Repeat (2d) for n = 4, m = 5, A = 9 10 11 12 13 14 16 15 17 18 20 19 b= 1243 5 and e=0.1.