I'm learning about Primal-Dual interior point algorithms from Boyd & Vandenberghe Convex Optimization, ch.11.7. Now the text mentions:
In a primal-dual interior-point method, the primal and dual iterates are not necessarily feasible.
However, when describing the algorithm more, it is stated:
Now consider the Newton step for solving the nonlinear equations rt(x, λ, ν) = 0, for fixed t (without first eliminating λ, as in §11.3.4), at a point (x, λ, ν) that satisifes f(x) ≺ 0, λ ≻ 0.
The text then describes the Newton step, and later on a modified backtracking line search to ensure that $\lambda$ stays positive / dual feasible. It reads to me like from this quote onwards we need our iterates to be strictly feasible...?
My question is simply: do I need an initial strictly feasible $x, \lambda$ with which to start my primal dual method? This seems to contradict the first quote.
For my particular problem it would be much more convenient to start with a not-necessarily feasible point. Also, my model of the inequality constraint functions changes at every iteration ideally (I am doing sequential convex optimization).
Hence I want an iterative method such that after every iteration I don't need to solve a phase I/II problem to get a feasible starting point (let alone strictly feasible?). Computation time is a major concern. Is the primal dual algorithm a good choice for me? Cheers.