Add tips on minimum-time problems (SleipnirGroup#707)

docs/tips.md

@@ -52,3 +52,63 @@ restriction. In other words, use `minimize(x ** 2 + y ** 2 + z ** 2)` instead of
 Store common subexpressions in intermediate variables and reuse them instead of
 writing out the subexpressions each time. This ensures common subexpressions in
 the expression tree are only traversed and updated once.
+
+## Minimum-time problems
+
+The obvious problem formulation for minimum-time problems uses one dt shared
+across all timesteps.
+```python
+import jormungandr as jmg
+
+N = 100
+T_max = 5.0
+
+problem = jmg.optimization.OptimizationProblem()
+
+x = problem.decision_variable(N + 1)
+v = problem.decision_variable(N)
+
+dt = problem.decision_variable()
+dt.set_value(T_max / N)
+problem.subject_to(dt > 0)
+problem.subject_to(dt < T_max / N)
+
+for k in range(N):
+    problem.subject_to(x[k + 1] == x[k] + v[k] * dt)
+
+problem.minimize(dt)
+
+problem.solve()
+```
+The nonzero initial value for dt avoids a degenerate case, and the upper bound
+prevents the solver exploiting discretization artifacts.
+
+This formulation can have feasibility issues though per section 15.3
+"Elimination of variables" of "Numerical Optimization, 2nd Ed.". Instead, we
+recommend using a separate dt for each timestep, with them all
+equality-constrained.
+```python
+import jormungandr as jmg
+
+N = 100
+T_max = 5.0
+
+problem = jmg.optimization.OptimizationProblem()
+
+x = problem.decision_variable(N + 1)
+v = problem.decision_variable(N)
+
+dt = problem.decision_variable(N)
+problem.subject_to(dt > 0)
+problem.subject_to(dt < T_max / N)
+for k in range(N - 1):
+    problem.subject_to(dt[k] == dt[k + 1])
+
+for k in range(N):
+    dt[k].set_value(T_max / N)
+    problem.subject_to(x[k + 1] == x[k] + v[k] * dt[k])
+
+problem.minimize(sum(dt))
+
+problem.solve()
+```