Comparison of relaxations

All problems were solved with IPOPT v.3.11 invoked from AMPL v.20130327 using the HSL linear algebra solvers. Most problems were run with the standard solver options, only the perspective formulation (5.41) needed a larger amount of iterations in combination with a slight modification of the initial parameters to converge for scenario 1. For the homotopy methods, IPOPTwarm start options were added.

The reference solutions were provided with a dynamic programming approach with a state discretization of 0.001 [m/s] for the velocity and 20 [Nm] for the engine torques Mindand

MEB. The fuel consumption is solely dependent on the velocity and gear choices and hence is not needed to be considered for the discretization. These solutions can be guaranteed to be sufficiently close to the global, optimal solution of the problem.

The following nine problems described in the previous section were solved for both scenarios: • IC of both the constraints and the ODE system,

• OC of both the constraints and the ODE system,

• relaxed vanishing constraints with OC of the ODE system, • smoothed, relaxed vanishing constraints with OC of the ODEs, • 0-corrected full perspective formulation,

• 0-corrected decomposed perspective formulation,

• tightened perspective formulation, i.e., smoothed, relaxed vanishing constraints with OC of ODE and lifted controls,

• dynamic programming to provide global, optimal solutions.

We present numerical results for two selected scenarios in Figures 5.14–5.17. The track’s height profile is shown on top, followed by nine plots of the relaxed (local) optimal solu- tions identified by IPOPTfor the described formulations and the global solution identified by BUCHNER’s dynamic programming code, cf. [42].

We use four properties to analyze the formulations’ behavior, which are the objective function value, the fractionality, the infeasibility and the computational time. The objective can be used to describe the tightness of the relaxation. The fractionality property is defined to be the average (over the different time steps) Manhattan norm difference from an integral point:

fractionality=def nXc−1 i=0 1 nc nµ X j=1 € 0.5 −yi, j− 0.5  Š_.

The infeasibility property describes the average infeasibility with regard to the constraints c(·) of the disjunction and is calculated using the vanishing constraint formulation (5.38). It provides the weighted violation of the constraints for each gear, the ones for inactive modes are set to 0 since their multipliers are 0. Here, aggregation effects – also called compensatory effects – can be seen best. The vanishing constraint interpretation of the constraints also enables feasible results after the usage of rounding strategies, which only round up nonzeros. For each control interval, the summed infinity norm violations are averaged:

infeasibilitydef = n_Xc−1 k=0 1 3 nc ‚ max 1¶i¶nµ ¦ yk,i € Mmax ind € neng,k,0,i Š − Mind,k Š© + max₁ ¶i¶nµ n yk,i 

neng,k,0,i− nmineng o + max₁ ¶i¶nµ n yk,i  nmax eng − neng,k,0,i o Œ .

The scenarios are chosen such that the deviation of the velocity profile is prioritized during optimization. The desired velocity is set to be constant 22 m/s. From both scenarios, a clear picture emerges.

• IC favors fractional solutions but they are computed very fast. Yet, they are quite far from satisfying the vanishing constraints. The solutions combine two gears to get a maximum acceleration maintaining feasibility with regard to the aggregated constraints – these aggregations allow compensatory effects to happen.

• OC is quite fast as well and and yields reasonable approximations, which also suffer from compensatory effects but to a much lesser degree than IC.

• The Big-M formulation of constraints with OC of the ODE lies somewhere in between the IC and OC with regard to the feasibility. However, it is farther away from integral solutions and also provides a weaker relaxation than both previous formulations. Since the computation times are comparable with OC, it is dominated by the OC formulation. • The relaxed vanishing constraint formulation succeeds in yielding feasible solutions for both scenarios. The fractionality is also very low. However, the effects of the non- convexity of the problem can be seen here very well as the provided solution is only a local solution. It is worse than the global solution found by dynamic programming. The neglect of global solutions by the usage of local solvers – as e.g. IPOPT– removes the relaxation property since only the global solution would provide a lower bound to the problem. For the computational time, around half of it is spent to solve the initial problem of the homotopy and the rest is spent driving " & 0 and re-solving the warm-started problems. In comparison to the previously presented formulations, the computational time is much higher.

• The relaxed and smoothed vanishing constraint formulation is quite comparable to the relaxed one but suffers even more from being stuck in a local solution while " & 0 and hence needs longer to compute.

• The full perspective formulation seems to be a very weak relaxation for this problem. The engine constraints can be satisfied while the highest and lowest gear are combined to allow maximum acceleration while the lowest gear is used as much as needed to hit the desired velocity profile.

• The slightly tighter decomposed variant of the perspective formulation provides a solu- tion that looks much better, but is still inferior in all aspects to the OC formulation. • The tightened perspective formulation needs much more computational time than the

other problems described so far. However, it provides the solution that is – with regard to the integral variables – the closest one to the optimal solution. As the other for- mulation with vanishing constraints, also this formulation succeeds in finding a locally optimal feasible point in one case. In the other case, it is a proper relaxation. The infea- sibility is due to the additional freedom of having lifted the controls. Their aggregation does not necessarily fulfill the constraints anymore.