Reconsider the example used to illustrate the interior-point algorithm in Sec. 8.4 of [HL] (you find the section in ItsLearning under Resources). Suppose that $(x1, x2) = (1, 3)$ were used instead as the initial feasible trial solution. Perform one or two iterations manually, starting from this solution. Then, write a python script to automatize the process. Finally solve this problem:
\[\begin{array}{rl} \text{maximize} \;\;&2x_1 + 3x_2 + 2x_3 \\ \text{subject to} \; \; &x_1 + x_2 +2x_3 = 3\\ &x_1,x_2,x_3 \geq 0. \end{array}\]using as starting solution $[x_1, x_2, x_3]=[1, 3/2, 1/4]$.
How should the algorithm change if the problem was a minimization problem?
Solve some of the problems from [KN1] with Gurobi and other solvers (soplex, glpsol) and analyze the logs.
Consider the Feature Selection case.
Implement the Lasso version together with minimizing the least absolute error. Solve the linear programming problem and analyze the solution process in the light of the article [KN1].
Implement the Lasso version (L1-norm as regularization component added to the objective function) together with minimizing the least absolute error instead of the square of residuals as done in the notebook.