Abstract One approach to solving optimal control problems is Bock’s direct multiple shoot-
ing method. This method yields lifted nonlinear optimization problems (NLPs) with a spe-
cific block structure. Exploiting this structure via tailored optimization algorithms can be
computationally beneficial. We propose such methods, primarily within the framework of fil-
ter line search sequential quadratic programming (SQP) methods and the solver blockSQP.
A key aspect introduced by multiple shooting is the distinction between free and dependent
variables. The dependent variables can be formally eliminated from the quadratic subprob-
lems (QPs) through a process known as condensing. We provide the theoretical background
for efficiently handling bounds on dependent variables by distinguishing between explicit
and implicit constraints. The latter arise from the interplay of bounds implicitly implied
by nonlinear dynamics and the continuity conditions of the lifted problem. Subsequently,
we analyze the iteration behavior of blockSQP using various optimal control examples and
present a tailored convexification strategy for indefinite QPs. Applying this strategy to con-
densed QPs yields further benefits in certain instances. Since scaling significantly influences
numerical optimization, we study its effect and discuss its connection to sizing strategies and
termination criteria. To make the optimization less sensitive to scaling, we introduce a termi-
nation scheme based on achieved progress and a scaling heuristic that balances derivatives
with respect to free and dependent variables. Numerical tests demonstrate that our proposed
methods offer advantages compared to both the initial version of blockSQP and the general-
purpose solver ipopt on a number of benchmark problems.