1. Consider the optimization problem
(1)subject to
(a) Determine if the problem if (1) is a convex problem. Justify your answer.
(b) 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) Verify that the reported optimal point (x∗,y∗) satisfies the constraint inequality and yields the optimal value.
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
f(x) ≤ t ⇐⇒ ϕt(x) ≤ 0
and ϕt is convex for all t.
(d) Let n = 3 and m = 3. Let . Use CVX and the bisection
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.
(e) Repeat (2d) for .
ines if any).
