1. Consider the optimization problem minimize
x∈R++ ,y∈R++ subject to
x2 y
x + y ≥ 1 x1 + y1 ≤ 1
(1)
x1.5 + y1.5 ≤ 51.5
(a) Determine if the problem if (1) is a convex problem. Justify your answer.
- (b) (10%) If Problem (1) is convex, use CVX to solve the problem. Write down the optimal value p∗ and an optimal point you obtained from CVX.
- (c) (10%) Verify that the reported optimal point (x∗,y∗) satisfies the constraint inequality and yields the optimal value.
- (d) (10%) Submit your source code as problem1.m or problem1.py.
2 Consider the optimization problem
minimize f0 (x) (2) x∈Rn
subject to ||Ax − b||1 ≤ ε where A ∈ Rm×n, b ∈ Rm, and ε > 0, and f0(x) = length(x).
- (a) (10%) Determine if Problem (2) is a convex problem. Justify your answer.
- (b) (10%) Show that Problem (2) is a quasiconvex problem. (Hint: You need to show that f0(x) is a
quasiconvex function in x.)
- (c) (10%) Construct a family of functions φt : Rn → R such that
and φt is convex for all t.
f(x) ≤ t ⇐⇒ φt(x) ≤ 0
123 1
(d) (10%)Letn=3andm=3. LetA= 4 5 6 ,b= 1 ,andε=0.1. UseCVXandthebisection
789 1
method your learned in class to solve the problem. Write down the optimal value p∗ and an optimal
point x∗ you obtained from your algorithm. Make sure that the reported optimal point x∗ satisfies the
constraint inequality and yields the optimal value.
1234 1
5 6 7 8 2
- (e) (10%)Repeat(2d)forn=4,m=5,A= 9 10 11 12 ,b= 4 ,andε=0.1. 13 14 16 15 3
17 18 20 19 5
- (f) (10%) Submit your source code as problem2.m or problem2.py (along with its subroutines if any).
