Question: For simplex algorithm we developed in class, does e_j stand for a vector with 1 on its j-th position and the rest zeros? or is it a vector of all ones?
Answer: 1 in its j-th position and the rest zeros.
Question: In simplex algorithm, at each iteration B loses one column and gains one. I know we remove the j-th column (corresponding to the most negative entry of multipliers). But what is the column we are adding? and what value do we put for x_j to make it non-zero?
Answer: The next point is x + alpha p, where p is determined from solving the system involving e_j, and alpha defines the longest step that is possible without violating any of the constraints x >= 0. The constraint that we hit becomes the added one.
Question: For IPM algorithm mentioned on page 582 of our book, what values of mu and theta should we use?
Answer: Your choice. Your book advises (Thm 17.1) a starting value for mu. Your book also suggests a value for theta (p. 582). Another choice that may need to be made is a step length parameter. Typically, algorithms step no closer than 90% or 99% of the way to the boundary x >= 0. If you omit the steplength, and don't adjust mu carefully, the algorithm may generate infeasible points.
Question: In description you mentioned "use 'qr' to solve the least squares problem at each iteration." What does that mean?
Answer: I mean that you should do a qr factorization of XA^T at each iteration and use the Q and R factors to solve problem a on p.582.