As we mentioned in the introduction of this section, the regularized decomposition algorithm works with a more general regularizing term of the form (a) Observe that the proof of convergence relies on...

As we mentioned in the introduction of this section, the regularized decomposition algorithm works with a more general regularizing term of the form

(a) Observe that the proof of convergence relies on strict convexity of the objective function (Lemma 5), thus α > 0 is needed. It also relies on
 which is simply obtained by taking a finite α . The algorithm can thus be tuned for any positive α and α can vary within the algorithm. (b) Taking the same starting point and data as in Exercise 2, show that by selecting different values of α , any point in ]−20,20] can be obtained as a solution of the regularized master at the second iteration (where 20 is the upper bound on x and the first iteration only consists of adding cuts on θ₁
and θ₂
).

(c) Again taking the same starting point and data as in Exercise 2, how would you take α to reduce the number of iterations? Discuss some alternatives.

(d) Let α = 1 for Iterations 1 and 2. As of Iteration 2, consider the following rule for changing α dynamically. For each null step, α is doubled. At each exact step, α is halved. Show why this would improve the performance of the regularized decomposition in the case of Exercise 2. Consider the starting point x1 = −0.5 as in Example 1 and observe that the same path as before is followed.