Feasibility issues in a primal-dual interior-point method for linear programming

Improved Full-Newton Step O(nL) Infeasible Interior-Point Method for Linear Optimization

“…We measure proximity to the μ-center of the perturbed problems by the quantity δ(x, s; μ) as defined in (4). Thus, initially we have δ(x, s; μ) = 0.…”

“…It differs from the classical IIPMs (e.g. [2][3][4][5][6][7][8][9]) in that the new method uses only full steps (instead of damped steps), which has the advantage that no line searches are needed. Our motivation for the use of full-Newton steps is that, though such methods are less greedy, the best complexity results for interior-point methods are obtained for such methods.…”

Implementation of interior point methods for mixed semidefinite and second order cone optimization problems

“…The interior point method should then generate either an approximate primal-dual optimal solution pair, or an approximate Farkas-type dual solution to certify that no (reasonably sized) feasible solution pair exists. In this section, we discuss two alternative approaches to deal with the cold start situation: the infeasible interior point method of Lustig [42,43] and the self-dual embedding technique of Ye, Todd and Mizuno [85].…”

“…This freedom was used by Lustig, Marsten and Shanno [43,44,46] to build highly effective interior point based solvers for linear programming. A further landmark was the introduction of the self-dual embedding technique by Ye et al [85,84], which provides a more elegant method for dealing with (in)feasibility issues than the infeasible interior point framework of Lustig [42]. A more complete (and probably less biased) overview is given by Freund and Mizuno [19].…”

The practical behavior of the homogeneous self-dual formulations in interior point methods

“…The main disadvantage of this approach are the necessary regularity conditions, for example the existence of a feasible primal and dual interior solution. As these conditions are often not practical in real-life, several techniques were developed to handle such degenerate cases [12,16,32].…”