I have an example which implements time-optimal control for Dubins vehicle. I have found that Nonconvex.jl is slower than JuMP. I think it is because Nonconvex.jl does not provide sparsity information to Ipopt. The nonzero entries in Jacobian and Hessian are much higher. Is there any way to get around this? I have added the code samples below for reference.

```
using Nonconvex
using Plots
Nonconvex.@load Ipopt
N = 40 # Number of knot points
Ns = 3 # Number of state variables
Nu = 1 # Number of Inputs
Nx = 1:N # state 1 indices
Ny = N+1:2N # state 2 indices
Nθ = 2N+1:3N # state 3 indices
Nu = 3N+1:4N # input indices
Nt = 4N+1 # time index
f(x) = x[Nt]
function g(x, i)
xs = x[Nx]
ys = x[Ny]
θs = x[Nθ]
u = x[Nu]
dt = x[Nt]
g1 = xs[i+1] - xs[i] - dt*cos(θs[i])
g2 = ys[i+1] - ys[i] - dt*sin(θs[i])
g3 = θs[i+1] - θs[i] - dt*u[i]
return [g1, g2, g3]
end
m = Model(f)
lb = -10*ones(Nt)
ub = 10*ones(Nt)
lb[Nu] .= -1.0
ub[Nu] .= 1.0
lb[Nt] = 2*π/N
ub[Nt] = 100/N
addvar!(m, lb, ub)
for i=1:N-1
add_eq_constraint!(m, x -> g(x, i))
end
function end_point(x)
xf = x[Nx]
yf = x[Ny]
return [xf[N]-0.5, yf[N]-0.5]
end
add_eq_constraint!(m, x -> end_point(x))
function initial_point(x)
x0 = x[Nx]
y0 = x[Ny]
θ0 = x[Nθ]
return [x0[1], y0[1]-0, θ0[1]-0]
end
add_eq_constraint!(m, x -> initial_point(x) )
alg = IpoptAlg()
options = IpoptOptions(first_order=false)
x0 = 0.1*ones(Nt)
result = optimize(m, alg, x0, options = options)
plot(result.minimizer[Nx], result.minimizer[Ny], aspect_ratio = 1)
```

The corresponding JuMP version:

```
using JuMP
import Ipopt
using Plots
# using LaTeXStrings
# Create JuMP model, using Ipopt as the solver
user_options = (
# "mu_strategy" => "monotone",
# "linear_solver" => "ma27",
)
dubins = Model(optimizer_with_attributes(Ipopt.Optimizer,user_options...))
#set_silent(dubins)
# Constants
vm = 1.0
um = 1.0 # Maximum thrust
x_f = 0.5
y_f = 0.5
n = 40 # Time steps
tmin = (2*pi*1)/vm
tmax = 100
# Decision variables
@variables(dubins, begin
(tmin)/n <= Δt <= (tmax)/n # Time step
## State variables
x[1:n] # Velocity
y[1:n] # Height
th[1:n] # Height
## Control variables
-um ≤ u[1:n] ≤ um # Thrust
end)
# Objective
@objective(dubins, Min, Δt)
## Initial conditions
fix(x[1], 0; force = true)
fix(y[1], 0; force = true)
fix(th[1], 0; force = true)
## Final conditions
fix(x[n], x_f; force = true)
fix(y[n], y_f; force = true)
# fix(th[n], pi/2; force = true)
## Dynamics
for j in 2:n
@NLconstraint(dubins, x[j] == x[j - 1] + Δt *0.5* vm*(cos(th[j - 1])+cos(th[j])))
## Trapezoidal integration
@NLconstraint(dubins, y[j] == y[j - 1] + Δt *0.5* vm*(sin(th[j - 1])+sin(th[j])))
## Trapezoidal integration
@NLconstraint(dubins, th[j] == th[j - 1] + 0.5*Δt * (u[j]+u[j-1]))
## Trapezoidal integration
end
# Solve for the control and state
println("Solving...")
optimize!(dubins)
solution_summary(dubins)
function plot_circle(xc, yc, rc, fig)
th = collect(0:0.1:2π)
plot!(fig, xc .+ rc*cos.(th), yc .+ rc*sin.(th), line =:dash, color =:red, label =nothing)
end
## Display results
println("Min time: ", objective_value(dubins)*n)
fig1 = plot(value.(x), value.(y),
label = "Vehicle trajectory",
aspect_ratio=:equal,
framestyle =:box, lw = 2)
```

Adding corresponding CasADi code for reference. Note that, in CasADi I can use explicit RK4 (or any other method) easily. I could not do that in JuMP. While it can be done in Nonconvex.jl there is the issue with sparsity.

```
using CasADi
using Plots
N = 40 # number of control intervals
x0 = [0; 0; 0] # Initial state
xf = [0.5; 0.5; pi / 2]
# Constant parameters
v = 1.0
opti = casadi.Opti()
# Decision variables [height;velocity;mass]
x = opti._variable(3, N)
xs = x[1,:]; ys =x[2,:]; ths = x[3,:]
u = opti._variable(1, N) # Control vector
dt = opti._variable(1, 1)
# Dynamic constraints
dubins_ode(x, u) = casadi.vertcat(cos(x[3]), sin(x[3]), u)
for k = 1:N-1
# Euler integration
# opti._subject_to(x[:,k+1] == casadi.plus(x[:,k], dt*dubins_ode(x[:,k], u[1,k])) )
# RK-4 Integration
k1 = dubins_ode(x[:,k] , u[:,k])
k2 = dubins_ode(casadi.plus(x[:,k] , dt*k1/2), u[:,k])
k3 = dubins_ode(casadi.plus(x[:,k] , dt*k2/2), u[:,k])
k4 = dubins_ode(casadi.plus(x[:,k] , dt*k3), u[:,k])
opti._subject_to(casadi.minus(x[:,k+1], casadi.plus(x[:,k], (dt/6)*(casadi.plus(casadi.plus(k1,k2),casadi.plus(k3,k4))))) == 0)
end
# Boundary conditions
opti._subject_to(casadi.minus(x[:,1], x0) == 0)
opti._subject_to(casadi.minus(x[1:2,end], xf[1:2]) == 0)
# Input constraints
opti._subject_to(u >= -1.0) # Bounded control
opti._subject_to(u <= 1.0) #
# Time constraints
mintime = 0.01/N
maxtime = 10/N
opti._subject_to(dt >= mintime)
opti._subject_to(dt <= maxtime)
# Objective
function obj(x, u, dt)
return dt
end
opti.minimize(obj(x, u, dt)) # minimize fuel consumption
# Initial Guess of solution
opti.set_initial(x, 10.0)
opti.set_initial(u, -1.0)
opti.set_initial(dt, 1.0/N)
# Solve
opti.solver("ipopt")
sol = opti.solve()
x = sol.value(x)
dt = sol.value(dt)
# Post-processing
t = LinRange(0, (N + 1) * sol.value(dt), N + 1)
plt1 = plot(x[1,:], x[2,:], aspect_ratio = 1)
```