这是一篇关于凸优化的Final Exam作业代写

1. Basic Knowledge
2. Is the set {a ∈ R k | p(0) = 1, |p(t)| ≤ 1 for α ≤ t ≤ β}, where p(t) = a1 + a2t + · · · + akt k−1 convex?

(10 points)

2. Prove that x ∗ = (1,1/2, −1) is optimal for the optimization problem

minimize 1/2x  T P x + q T x + r

subject to − 1 ≤ xi ≤ 1, i = 1, 2, 3,

where

P = {13 12 −2

12 17 6

−2 6 12}

, q = {−  22

− 14.5

13}, r = 1.

(10 points)

3. Equivalently reformulate the following problem into a standard convex optimization form, i.e., Linear Programming (LP), Second-Order Cone Programming (SOCP) and Semidefifinite Programming(SDP).

Minimize ||Ax − b||∞

(10 points)

1. KKT Conditions

Let n ≥ 2. Consider the following minimization problem over x = [x1, . . . , xn] T ,

minx nXj=1cj xj

s.t.nXj=1xj = 1,

xj ≥ , j = 1, . . . , n,

where cj > 0, ∀j,  > 0 are parameters.

(1) Determine whether this problem is convex or not, and provide your argument. (5 points)

(2) Write down the dual problem of (1). (5 points)

(3) Derive the KKT conditions of (1). (5 points)

(4) Derive the expression of the optimal solution of (1). (10 points)

III. Gradient Methods

1. Let f : R d → R be a convex and difffferentiable function, it is said to be M-smooth if

f(y) ≤ f(x) + h∇f(x), y − xi +M 2k y − xk 2 2 , ∀x, y ∈ R d .  (2)

Prove that the above M-smooth condition holds if and only if  k∇f(x) − ∇f(y)k 2 ≤ Mk x − yk 2,  ∀x, y ∈ R d .  (3)

(20 points)

2. The subgradient method updates are given by x ` +1 = x ` − α ` g ` , where g ` ∈ ∂f(x ` ). Suppose f is convex with fifinite unconstrained minimum f ? , and is L-Lipschitz. Letting X ? denote the space of minimizers (taken to be non-empty and closed), defifine the distance function distX ? (u) =infz∈X ? k z − uk 2, and suppose there exists some µ > 0 such that distX ? (y) ≤1µ (f(y) − f ? ) for all y ∈ dom(f). Prove that dist2 X ? (x ` +1) ≤  1 − µ 2 L 2  ` +1 dist2 X ? (x 0 ).

(15 points)

IV.Proximal Algorithm

For 0 ≺ α, β ≺ 1, defifine hα,β : R n → R as

hα,β(x) = α T x+ + β T x−,

where x+ = max{x, 0} and x− = max{−x, 0}, the maximum is taken elementwise.

Give a simple expression for the proximal operator of hα,β. (10 points)