Midpoint vs Trapeze

Note

The linear solver is MUMPS for CPU experiments and CUDSS for GPU experiments.
Below you can find Dolan–Moré performance profiles comparing discretization method–solver combinations on the set of optimal control problems. For a detailed explanation of how to read these profiles, see the Performance Profiles page.

Comparison of Discretization Methods

This benchmark evaluates the effect of two discretization methods:

Midpoint
Trapeze

combined with two nonlinear solvers:

Ipopt
MadNLP

on the ExaModels optimal control modeler.

The comparison is based on four combinations (disc_method, solver):

(trapeze, ipopt)
(trapeze, madnlp)
(midpoint, ipopt)
(midpoint, madnlp)

⚙️ Configuration

Problems: balanced_field, beam, brachistochrone, bryson_denham, chain, dielectrophoretic_particle, double_oscillator, ducted_fan, electric_vehicle, glider, insurance, jackson, mountain_car, robbins, robot, rocket, space_shuttle, steering, vanderpol
Solvers: ipopt, madnlp
Models: exa
Grid sizes: 200 discretization points
Discretization: trapeze, midpoint method
Tolerance: 1.0e-8
Ipopt strategy: adaptive barrier parameter
Limits: 2000 iterations max, 600.0s wall time

🖥️ Environment

📅 Timestamp     : 2026-03-10 19:11:59 UTC
🔧 Julia version : 1.12.5
💻 OS            : Linux
🖥️ Machine       : runnervm0kj6c

You can download the exact environment used for this benchmark:

📦 Project.toml - Package dependencies
📋 Manifest.toml - Complete dependency tree with versions
📜 Benchmark script - Julia script to run the benchmark

These files allow you to reproduce the benchmark environment and results exactly.

ℹ️ Version info

Julia Version 1.12.5
Commit 5fe89b8ddc1 (2026-02-09 16:05 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver3)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager

📦 Package status

Project CTBenchmarks v0.3.1
Status `~/work/CTBenchmarks.jl/CTBenchmarks.jl/Project.toml`
  [6e4b80f9] BenchmarkTools v1.6.3
⌅ [54762871] CTBase v0.16.2
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  [682c06a0] JSON v1.4.0
  [4076af6c] JuMP v1.30.0
⌅ [d72a61cc] MadNLPGPU v0.7.18
  [3b83494e] MadNLPMumps v0.5.1
  [f4238b75] NLPModelsIpopt v0.11.2
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  [6462fe0b] Sockets v1.11.0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`

📚 Complete manifest

Project CTBenchmarks v0.3.1
Status `~/work/CTBenchmarks.jl/CTBenchmarks.jl/Manifest.toml`
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`

📈 Performance Profile: CPU Time

Performance Profile Analysis

Dataset overview for core-midpoint-trapeze:

Problems: 19 unique optimal control problems
Instances: 19
Solver combos: 4

Profile configuration:

Instance definition: (problem, grid_size)
Solver combos definition: (disc_method, solver)
Criterion: CPU time
Successful runs: 76/76 (100.0%)
Successful instances: 19/19 (100.0%)
Unsuccessful instances: none (every instance had at least one successful run)

Robustness (% of instances solved):

(trapeze, madnlp): 100.0%
(midpoint, madnlp): 100.0%
(trapeze, ipopt): 100.0%
(midpoint, ipopt): 100.0%

Efficiency (% of instances where fastest):

(trapeze, madnlp): 36.8%
(midpoint, madnlp): 57.9%
(trapeze, ipopt): 0.0%
(midpoint, ipopt): 5.3%

Most robust: 4 combinations tied at 100.0%.

Most efficient: (midpoint, madnlp) was fastest on 57.9% of instances.

📊 Result Tables

Problem:

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	38.693	17	771.011400	min
✓	200	`exa`	`ipopt`	39.517	19	771.009269	min
✓	200	`exa`	`madnlp`	39.576	23	771.011400	min
✓	200	`exa`	`madnlp`	35.459	23	771.009269	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	15.385	16	8.891335	min
✓	200	`exa`	`ipopt`	13.688	15	8.889505	min
✓	200	`exa`	`madnlp`	12.026	21	8.891335	min	✓
✓	200	`exa`	`madnlp`	15.034	23	8.889505	min

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	30.834	19	1.802935	min
✓	200	`exa`	`ipopt`	27.339	19	1.802937	min
✓	200	`exa`	`madnlp`	23.718	20	1.802935	min
✓	200	`exa`	`madnlp`	22.065	20	1.802937	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	12.450	13	4.000889	min
✓	200	`exa`	`ipopt`	11.768	13	4.000222	min
✓	200	`exa`	`madnlp`	8.606	15	4.000889	min	✓
✓	200	`exa`	`madnlp`	9.666	15	4.000222	min

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	9.618	7	5.068917	min
✓	200	`exa`	`ipopt`	13.676	12	5.068486	min
✓	200	`exa`	`madnlp`	8.091	7	5.068917	min	✓
✓	200	`exa`	`madnlp`	11.629	12	5.068486	min

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	13.032	12	7.811387	min
✓	200	`exa`	`ipopt`	13.401	13	7.811390	min
✓	200	`exa`	`madnlp`	9.163	13	7.811387	min	✓
✓	200	`exa`	`madnlp`	9.374	14	7.811390	min

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	11.214	5	0.000910	min
✓	200	`exa`	`ipopt`	9.673	5	0.000910	min
✓	200	`exa`	`madnlp`	8.216	5	0.000910	min
✓	200	`exa`	`madnlp`	7.183	5	0.000910	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	487.905	92	1831.941573	min
✓	200	`exa`	`ipopt`	390.539	91	1831.889795	min
✓	200	`exa`	`madnlp`	154.004	41	1831.941575	min
✓	200	`exa`	`madnlp`	140.925	48	1831.889803	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	4.549	4	1228.698880	min
✓	200	`exa`	`ipopt`	4.365	4	1228.639163	min
✓	200	`exa`	`madnlp`	3.665	4	1228.698880	min	✓
✓	200	`exa`	`madnlp`	3.832	4	1228.639163	min

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	324.474	78	-1248.896363	min
✓	200	`exa`	`ipopt`	118.511	38	-1247.918471	min
✓	200	`exa`	`madnlp`	318.623	112	-1248.896363	min
✓	200	`exa`	`madnlp`	71.498	30	-1247.918471	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	340.853	91	-2.058581	min
✓	200	`exa`	`ipopt`	3408.557	143	-2.057424	min
✓	200	`exa`	`madnlp`	340.188	120	-2.058581	min	✓
✓	200	`exa`	`madnlp`	369.592	124	-2.057424	min

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	81.991	42	-0.191861	min
✓	200	`exa`	`ipopt`	87.101	48	-0.191840	min
✓	200	`exa`	`madnlp`	49.681	40	-0.191861	min
✓	200	`exa`	`madnlp`	37.486	33	-0.191840	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	63.619	41	102.397570	min
✓	200	`exa`	`ipopt`	53.776	39	102.384808	min
✓	200	`exa`	`madnlp`	53.628	49	102.380995	min
✓	200	`exa`	`madnlp`	34.204	39	102.384808	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	23.580	19	19.626260	min
✓	200	`exa`	`ipopt`	21.396	19	19.459465	min	✓
✓	200	`exa`	`madnlp`	26.868	34	19.626260	min
✓	200	`exa`	`madnlp`	23.565	19	19.459465	min

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	65.674	18	9.141395	min
✓	200	`exa`	`ipopt`	52.621	15	9.140867	min
✓	200	`exa`	`madnlp`	49.394	16	9.141397	min
✓	200	`exa`	`madnlp`	45.145	16	9.140868	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	28.546	18	-1.012836	min
✓	200	`exa`	`ipopt`	25.799	17	-1.012838	min
✓	200	`exa`	`madnlp`	22.603	21	-1.012835	min
✓	200	`exa`	`madnlp`	22.132	22	-1.012837	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	722.335	101	-0.595869	min
✓	200	`exa`	`ipopt`	708.524	106	-0.595876	min
✓	200	`exa`	`madnlp`	938.387	181	-0.595869	min
✓	200	`exa`	`madnlp`	332.089	74	-0.595876	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	18.921	8	0.554577	min
✓	200	`exa`	`ipopt`	18.292	8	0.554572	min
✓	200	`exa`	`madnlp`	17.811	11	0.554577	min
✓	200	`exa`	`madnlp`	16.099	10	0.554572	min	✓

Success	N	Model	Solver	Time (ms)	Iters	Objective	Criterion	Best
✓	200	`exa`	`ipopt`	3.680	3	1.047836	min
✓	200	`exa`	`ipopt`	3.897	3	1.047802	min
✓	200	`exa`	`madnlp`	2.817	3	1.047836	min	✓
✓	200	`exa`	`madnlp`	3.838	3	1.047802	min

ℹ️ Log

Benchmarks results:

┌─ Problem: balanced_field
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  39.576 ms | iters: 23    | obj: 7.710114e+02  (min) | CPU:   5.62 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  35.459 ms | iters: 23    | obj: 7.710093e+02  (min) | CPU:   5.32 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  38.693 ms | iters: 17    | obj: 7.710114e+02  (min) | CPU: 129.11 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  39.517 ms | iters: 19    | obj: 7.710093e+02  (min) | CPU: 127.05 KiB
│  └─
└─

┌─ Problem: beam
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  12.026 ms | iters: 21    | obj: 8.891335e+00  (min) | CPU:   1.76 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  15.034 ms | iters: 23    | obj: 8.889505e+00  (min) | CPU:   1.79 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  15.385 ms | iters: 16    | obj: 8.891335e+00  (min) | CPU:  72.55 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  13.688 ms | iters: 15    | obj: 8.889505e+00  (min) | CPU:  70.39 KiB
│  └─
└─

┌─ Problem: brachistochrone
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  23.718 ms | iters: 20    | obj: 1.802935e+00  (min) | CPU:   2.88 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  22.065 ms | iters: 20    | obj: 1.802937e+00  (min) | CPU:   2.55 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  30.834 ms | iters: 19    | obj: 1.802935e+00  (min) | CPU:  96.05 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  27.339 ms | iters: 19    | obj: 1.802937e+00  (min) | CPU:  95.20 KiB
│  └─
└─

┌─ Problem: bryson_denham
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   8.606 ms | iters: 15    | obj: 4.000889e+00  (min) | CPU:   1.33 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   9.666 ms | iters: 15    | obj: 4.000222e+00  (min) | CPU:   1.26 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  12.450 ms | iters: 13    | obj: 4.000889e+00  (min) | CPU:  69.45 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  11.768 ms | iters: 13    | obj: 4.000222e+00  (min) | CPU:  68.33 KiB
│  └─
└─

┌─ Problem: chain
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   8.091 ms | iters: 7     | obj: 5.068917e+00  (min) | CPU:   1.43 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  11.629 ms | iters: 12    | obj: 5.068486e+00  (min) | CPU:   1.37 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   9.618 ms | iters: 7     | obj: 5.068917e+00  (min) | CPU:  70.12 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  13.676 ms | iters: 12    | obj: 5.068486e+00  (min) | CPU:  72.94 KiB
│  └─
└─

┌─ Problem: dielectrophoretic_particle
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   9.163 ms | iters: 13    | obj: 7.811387e+00  (min) | CPU:   1.95 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   9.374 ms | iters: 14    | obj: 7.811390e+00  (min) | CPU:   1.75 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  13.032 ms | iters: 12    | obj: 7.811387e+00  (min) | CPU:  72.77 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  13.401 ms | iters: 13    | obj: 7.811390e+00  (min) | CPU:  72.48 KiB
│  └─
└─

┌─ Problem: double_oscillator
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   8.216 ms | iters: 5     | obj: 9.103362e-04  (min) | CPU:   2.71 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   7.183 ms | iters: 5     | obj: 9.101904e-04  (min) | CPU:   2.11 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  11.214 ms | iters: 5     | obj: 9.103362e-04  (min) | CPU: 102.86 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   9.673 ms | iters: 5     | obj: 9.101904e-04  (min) | CPU: 100.23 KiB
│  └─
└─

┌─ Problem: ducted_fan
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 154.004 ms | iters: 41    | obj: 1.831942e+03  (min) | CPU:   9.51 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 140.925 ms | iters: 48    | obj: 1.831890e+03  (min) | CPU:  10.14 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 487.905 ms | iters: 92    | obj: 1.831942e+03  (min) | CPU: 254.80 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 390.539 ms | iters: 91    | obj: 1.831890e+03  (min) | CPU: 252.38 KiB
│  └─
└─

┌─ Problem: electric_vehicle
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   3.665 ms | iters: 4     | obj: 1.228699e+03  (min) | CPU:   1.33 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   3.832 ms | iters: 4     | obj: 1.228639e+03  (min) | CPU:   1.03 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   4.549 ms | iters: 4     | obj: 1.228699e+03  (min) | CPU:  58.38 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   4.365 ms | iters: 4     | obj: 1.228639e+03  (min) | CPU:  54.53 KiB
│  └─
└─

┌─ Problem: glider
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 318.623 ms | iters: 112   | obj: -1.248896e+03 (min) | CPU:  14.30 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  71.498 ms | iters: 30    | obj: -1.247918e+03 (min) | CPU:   6.57 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 324.474 ms | iters: 78    | obj: -1.248896e+03 (min) | CPU: 268.72 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 118.511 ms | iters: 38    | obj: -1.247918e+03 (min) | CPU: 219.34 KiB
│  └─
└─

┌─ Problem: insurance
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 340.188 ms | iters: 120   | obj: -2.058581e+00 (min) | CPU:  26.32 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 369.592 ms | iters: 124   | obj: -2.057424e+00 (min) | CPU:  24.74 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 340.853 ms | iters: 91    | obj: -2.058581e+00 (min) | CPU: 246.09 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   3.409 s  | iters: 143   | obj: -2.057424e+00 (min) | CPU: 297.06 KiB
│  └─
└─

┌─ Problem: jackson
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  49.681 ms | iters: 40    | obj: -1.918610e-01 (min) | CPU:   6.74 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  37.486 ms | iters: 33    | obj: -1.918402e-01 (min) | CPU:   5.55 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  81.991 ms | iters: 42    | obj: -1.918613e-01 (min) | CPU: 124.73 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  87.101 ms | iters: 48    | obj: -1.918405e-01 (min) | CPU: 126.94 KiB
│  └─
└─

┌─ Problem: mountain_car
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  53.628 ms | iters: 49    | obj: 1.023810e+02  (min) | CPU:   5.51 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  34.204 ms | iters: 39    | obj: 1.023848e+02  (min) | CPU:   4.48 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  63.619 ms | iters: 41    | obj: 1.023976e+02  (min) | CPU: 100.95 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  53.776 ms | iters: 39    | obj: 1.023848e+02  (min) | CPU:  97.86 KiB
│  └─
└─

┌─ Problem: robbins
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  26.868 ms | iters: 34    | obj: 1.962626e+01  (min) | CPU:   2.65 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  23.565 ms | iters: 19    | obj: 1.945946e+01  (min) | CPU:   1.92 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  23.580 ms | iters: 19    | obj: 1.962626e+01  (min) | CPU:  95.61 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  21.396 ms | iters: 19    | obj: 1.945946e+01  (min) | CPU:  94.50 KiB
│  └─
└─

┌─ Problem: robot
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  49.394 ms | iters: 16    | obj: 9.141397e+00  (min) | CPU:   6.61 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  45.145 ms | iters: 16    | obj: 9.140868e+00  (min) | CPU:   6.25 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  65.674 ms | iters: 18    | obj: 9.141395e+00  (min) | CPU: 181.27 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  52.621 ms | iters: 15    | obj: 9.140867e+00  (min) | CPU: 175.55 KiB
│  └─
└─

┌─ Problem: rocket
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  22.603 ms | iters: 21    | obj: -1.012835e+00 (min) | CPU:   4.41 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  22.132 ms | iters: 22    | obj: -1.012837e+00 (min) | CPU:   4.27 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  28.546 ms | iters: 18    | obj: -1.012836e+00 (min) | CPU: 101.02 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  25.799 ms | iters: 17    | obj: -1.012838e+00 (min) | CPU:  98.86 KiB
│  └─
└─

┌─ Problem: space_shuttle
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 938.387 ms | iters: 181   | obj: -5.958694e-01 (min) | CPU:  52.66 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 332.089 ms | iters: 74    | obj: -5.958757e-01 (min) | CPU:  24.64 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 722.335 ms | iters: 101   | obj: -5.958694e-01 (min) | CPU: 309.17 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time: 708.524 ms | iters: 106   | obj: -5.958757e-01 (min) | CPU: 305.33 KiB
│  └─
└─

┌─ Problem: steering
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  17.811 ms | iters: 11    | obj: 5.545770e-01  (min) | CPU:   2.35 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  16.099 ms | iters: 10    | obj: 5.545724e-01  (min) | CPU:   2.12 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  18.921 ms | iters: 8     | obj: 5.545770e-01  (min) | CPU: 102.95 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:  18.292 ms | iters: 8     | obj: 5.545724e-01  (min) | CPU: 101.83 KiB
│  └─
└─

┌─ Problem: vanderpol
│
├──┬ Solver: madnlp, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   2.817 ms | iters: 3     | obj: 1.047836e+00  (min) | CPU:   1.33 MiB
│  └─
│
├──┬ Solver: madnlp, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   3.838 ms | iters: 3     | obj: 1.047802e+00  (min) | CPU:   1.08 MiB
│  └─
│
├──┬ Solver: ipopt, Discretization: trapeze
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   3.680 ms | iters: 3     | obj: 1.047836e+00  (min) | CPU:  53.50 KiB
│  └─
│
├──┬ Solver: ipopt, Discretization: midpoint
│  │
│  │  N = 200
│  │  ✓ | exa      | time:   3.897 ms | iters: 3     | obj: 1.047802e+00  (min) | CPU:  52.19 KiB
│  └─
└─