  We analyze sequences generated by interior point methods (IPMs) in convex and
nonconvex settings. We prove that moving the primal feasibility at the same
rate as the barrier parameter $\mu$ ensures the Lagrange multiplier sequence
remains bounded, provided the limit point of the primal sequence has a Lagrange
multiplier. This result does not require constraint qualifications. We also
guarantee the IPM finds a solution satisfying strict complementarity if one
exists. On the other hand, if the primal feasibility is reduced too slowly,
then the algorithm converges to a point of minimal complementarity; if the
primal feasibility is reduced too quickly and the set of Lagrange multipliers
is unbounded, then the norm of the Lagrange multiplier tends to infinity.
  Our theory has important implications for the design of IPMs. Specifically,
we show that IPOPT, an algorithm that does not carefully control primal
feasibility has practical issues with the dual multipliers values growing to
unnecessarily large values. Conversely, the one-phase IPM of
\citet*{hinder2018one}, an algorithm that controls primal feasibility as our
theory suggests, has no such issue.
