Reference no: EM132804665

Problem - Elastic net regularization

In statistics, the Elastic net uses a regularization that linearly combines the L1 and L2 penalties. The estimates from the elastic net method can be denoted as

β^{^} = arg min_β(f(β) + λ₁||β|| + λ₂||β||₂²

where f(β) is the loss function without regularization subject to the optimization variable β and λ₁, λ₂ are the coefficients to control the magnitude of penalty. Now we consider a linear model with

f(β) = ||y - xβ||₂²

where y ∈ Rⁿ X∈R^nxp, and β∈R^P. Please find the data of X and y with in = 500 and p = 200 via Canvas, solve the problem with different values of (A1, A2), then fill in the table by computing

Σ^p_i=1 I(|β_i*|≤ ∈)

where β* is the corresponding optimal solution, ∈ is set to be 10^-3, and the indicator function is defined as

I(|β_i*|≤ ∈) = 1   if |β_i*| ≤ ∈,

                   0      otherwise.

(b) Solve the following optimization problem:

minimize λmax(X) - λmin(X)
x∈s₊ⁿ
subject to trace (AX) = 1.

Here we consider

Please show the optimal value of X and all the code you use to solve the problem.

(c) Solve the unconstrained optimization problem with piecewise-linear objective function

minimize Max_i{a_ix ± b_i}
x∈R

where a = (a1, a2, a3, a4, a5) = (-2, -4, -8,1, 3) and b = (b1, b2, b3, b4, b5) = (-1, -6, -1, -1, -12). Compare the results with the solution of following optimization problem:

minimize t
xER
subject to t ≥ aix + bi, i = 1,...,5.

Please show the optimal value of x for both problems and all the code you use to solve the problem.