A polynomial path following algorithm for convex programming

A polynomial time infeasible interior-point arc-search algorithm for convex optimization

“…, p}, either zi or si is positive. Thus, if zi > 0, (A3) implies si = 0, therefore we know di = 0 due to (9); Similarly, if si > 0, (A3) implies zi = 0, we know ĉi = 0 due to (9). From the second and third rows of (8), we have…”

“…In view of (A2), we conclude â = 0. Then, it follows from (10) that d = 0, therefore, we know si ĉi = 0 for each i from (9). If si > 0, it holds ĉi = 0 for i / ∈ I(x).…”

“…In this paper, we extend the arc-search techniques to the convex nonlinear optimization problem for which various line-search interior point algorithms have been developed in [1,3,5,6,9,10,11,15,21], because many application problems can be formulated as a convex nonlinear optimization problem [4,17,24]. Although a polynomial bound has been proved for a feasible interior-point algorithm for the convex nonlinear optimization problem [15], to our best knowledge, there is no polynomial bound for infeasible interior-point algorithms for the convex nonlinear optimization problem because the latter is much more difficult [25].…”