Have a trajectory optimization problem posed as NLP. How to solve it with Gekko? Currently facing "Exception: @error: Max Equation Length" error

Question

I have a trajectory optimization problem posed as a Non linear program. The states x and controls u for each time step are concatenated into a single vector. By using indexing to extract the appropriate x's and u's, I want to add constraints for dynamics and also create an objective function which is the sum of x'Qx +u'Ru for each time step.I also have an initial condition and goal condition to be set as constraints

At the moment, the following is my code where I have created an array Z but to enforce the initial and goal conditions, I had to go element by element. Even though it works,is there a better way to do this?

nx = 13
nu = 4
N = int(self.tf/self.h)+1
         
Q = np.diag(np.ones(nx))
R = 0.1*np.diag(np.ones(nu))
Qf = 10*np.diag(np.ones(nx))    

#Initialize Model
m = GEKKO(remote=False)
#Set global options
m.options.IMODE = 3 #steady state optimization   
m.options.SOLVER = 3             
Z = m.Array(m.Var,self.nz,value=0,lb=-1e5,ub=1e5)

#Set upper and lower bounds: 1) U_lb = 0 
for i in np.arange(0,N-1):
    for j in self.u_index[:,i]:
        Z[j].lower = 0

for count, ele in enumerate(self.x_index[:,0]):
        print(Z[ele],xic[count])
        m.Equation(Z[ele]==xic[count])

for count, ele in enumerate(self.x_index[:,N-1]):
        print(Z[ele],xgoal[count])
        m.Equation(Z[ele]==xgoal[count])

#Objective
J = []
for i in np.arange(0,N-1):
        x = Z[self.x_index[:,i]]
        u = Z[self.u_index[:,i]]
        J.append(0.5*(x-xgoal).T@Q@(x-xgoal) + 0.5*u.T@R@u)
        
#Solve simulation          
m.Minimize(J)     
m.solve()

My cost function is currently the main issue as it causes the following error: Exception: @error: Max Equation Length" error. Any suggestions on how to tackle this? I am aware there is another method of solving optimal control problems using Gekko but I would like to try to solve it as an NLP

Answer 1

The current code should include a summation for the values of J.

for i in np.arange(0,N-1):
        x = Z[self.x_index[:,i]]
        u = Z[self.u_index[:,i]]
        J.append(0.5*(x-xgoal).T@Q@(x-xgoal) + 0.5*u.T@R@u)
        
#Solve simulation          
m.Minimize(m.sum(J))

Alternatively, add each objective term in the loop without defining J.

for i in np.arange(0,N-1):
        x = Z[self.x_index[:,i]]
        u = Z[self.u_index[:,i]]
        m.Minimize(0.5*(x-xgoal).T@Q@(x-xgoal) + 0.5*u.T@R@u)

The maximum equation size in Gekko is 15,000 characters as shown in gk0_model.apm file with m.open_folder() to inspect that model text file. The alternative summation should help break up each symbolic expression into smaller pieces.

If this doesn't solve the problem, consider using sparse matrix operations in Gekko such as the axb and qobj functions.