Easy NLP modeling in CasADi with Opti

Release 3.3.0 of CasADi introduced a compact syntax for NLP modeling, using a set of helper classes, collectively known as ‘Opti stack’.

In this post, we briefly demonstrates this functionality.

Rosenbrock problem

Let’s consider the classic Rosenbrock problem to begin with:

$$ \begin{align} \displaystyle \underset{x,y} {\text{minimize}}\quad &\displaystyle (1-x)^2+(y-x^2)^2 \ \end{align} $$

In CasADi’s Opti syntax, we can easily transcribe this to computer code:

opti = casadi.Opti();

x = opti.variable();
y = opti.variable();

opti.minimize((1-x)^2+(y-x^2)^2);

opti.solver('ipopt');
sol = opti.solve();

plot(sol.value(x),sol.value(y),'o');

Let’s make a variation on the problem by adding an equality constraint:

$$ \begin{align} \displaystyle \underset{x,y} {\text{minimize}}\quad &\displaystyle (1-x)^2+(y-x^2)^2 \newline \text{subject to} \, \quad & x^2+y^2=1 \end{align} $$

opti.minimize((1-x)^2+(y-x^2)^2);
opti.subject_to(x^2+y^2==1);

opti.solver('ipopt');
sol = opti.solve();

We may add in fact any number of equality/inequality constraints:

$$ \begin{align} \displaystyle \underset{x,y} {\text{minimize}}\quad &\displaystyle (1-x)^2+(y-x^2)^2 \newline \text{subject to} \, \quad & x^2+y^2=1 \newline & y\geq x \end{align} $$

opti.minimize((1-x)^2+(y-x^2)^2);
opti.subject_to(x^2+y^2==1);
opti.subject_to(y>=x);

opti.solver('ipopt');
sol = opti.solve();

We can also create a parametric NLP, were a parameter is getting fixed at solution time:

opti = casadi.Opti();

x = opti.variable();
y = opti.variable();
r = opti.parameter();

opti.minimize((1-x)^2+(y-x^2)^2);
con = x^2+y^2<=r;
opti.subject_to(con);
opti.solver('ipopt');

for r_value=linspace(1,3,25)
    opti.set_value(r,r_value)
    sol = opti.solve();
    plot(r_value,sol.value(f),'ko')
end

We may access the value of the Lagrange multiplier associated with the constraint using:

sol.value(opti.dual(con))

The Lagrange multiplier can be interpreted as the sensitivity of the optimal cost with respect to the relaxation of the constraint. We plotted the slope given by the Langrange multiplier in red in the figure below.

Download code: rosenbrock.m

Hanging chain problem

Next, we will visit the hanging chain problem. We consider $N$ point masses, connected by springs, hung from two fixed points at $(-2,1)$ and $(2,1)$, subject to gravity.

We seek the rest position of the system, obtained by minimizing the total energy in the system.

Consider that point mass $i$ has position $(x_i,y_i)$, we can write the gravitational potential energy as

$$ V_g = g m \sum_{i=1}^N y_i, $$

and the total spring potential energy as:

$$ V_s = \frac{1}{2} \sum_{i=1}^{N-1} D \left((x_i-x_{i+1})^2+(y_i-y_{i+1})^2\right). $$

We can do this in Opti using a oneliner, or a for loop:

opti = casadi.Opti();

p = opti.variable(2,N);
x = p(1,:);
y = p(2,:);

V = 0.5*D*sum((x(1:N-1)-x(2:N)).^2+(y(1:N-1)-y(2:N)).^2);
V = V + g*sum(m*y);

opti.minimize(V);

opti.subject_to(p(:,1)==[-2;1])
opti.subject_to(p(:,end)==[2;1])

opti.solver('ipopt');
sol = opti.solve();

plot(sol.value(x),sol.value(y),'-o')

After a first solve, further constraints can be added, e.g. a ground constrained

opti.subject_to(y>=cos(0.1*x)-0.5);
sol = opti.solve();

We can make the problem more numerically challenging by setting a nonzero restlength for the spring:

V = 0.5*sum(D*(sqrt((x(1:N-1)-x(2:N)).^2+(y(1:N-1)-y(2:N)).^2)-L/N).^2);
V = V + g*sum(m*y);

opti.minimize(V);

When the problem is nonconvex it’s always a good idea to provide initial guesses for your decision variables:

opti.set_initial(x,linspace(-2,2,N));
opti.set_initial(y,1);

When the going get’s tough, you may find it helpful to plot the intermediate solution for each iteration of the solver:

opti.callback(@(i) plot(opti.debug.value(x),opti.debug.value(y),'DisplayName',num2str(i)))

(Credits to Milan Vukov for delivering this problem)

Download code: chain.m

For more details about Opti, see Chapter 9 of the users guide. For an optimal control example, see the race car example.

Using Opti, you have the scalability of CasADi algorithmic differentiation available, in a user friendly packaging. Ideal for teaching!