Solving trajectory optimization problems via nonlinear programming: the brachistochrone case study

Abstract

This note discusses reformulations the brachistochrone problem suitable for solution via NLP. The availability of solvers and modeling languages such as AMPL (Fourer et al., AMPL: a modeling language for mathematical programming, 2003) makes it tempting to formulate discretized optimization problems and get solutions to the discretized versions of trajectory optimization problems. We use the famous brachistochrone problem to warn that the resulting solutions may be far different from the true optimal trajectory. Actually, we use our knowledge of the brachistochrone to argue that without this knowledge, for this particular example, we could not distinguish the true solution (a cycloid) from spurious solutions obtained by a natural discretization.

Appendix

The model (1) and (2) are directly formulated using AMPL. We present the model (2) with the differences between both models commented out.

After simplifications, the AMPL system reduces the problem (2) to 255 variables and 129 nonlinear equality constraints.

For the formulation (1), we rather have 509 variables and 383 equality constraints. I would have expected that AMPL detects the substitution to get rid of the Nx , Ny variables in model (1).

We present the number of iterations required for the NEOS solvers using all default options (we simply submitted the models on the NEOS server) for 64 and 128 discretization points for both models. This is not intended as a detailed benchmark, but just an indication on the potential difficulty of the models. Moreover, notice that there are only equality constraints, so that the interior point approaches and the SQP ones should be on par, no inequality constraint being present to distinguish them. Instances labeled as $\infty$ denote a reported failure due to reaching the default maximum number of iterations or function evaluations. × in the MINOS case refer to abortion (the message is “ MINOS 5.51: solution aborted .”). The R is caused by failure in the restoration phase in IPOPT. In some cases, the solution at hand looked close to be optimal/feasible. Other versions of LOQO appear to be able to solve the models. We do not explore more deeply those statistics, our point being to illustrate the potential difference on the difficulty to solve mathematically equivalent models. Indeed, although mathematically equivalent, model (1) has twice as many variables as (2) but solvers are much more challenged to solve the smaller model (Table 1).

Table 1 Number of iterations for models (1) and (2) using 64 and 128 discretization points