Lagrangian as symbolic object

[Markus Svedberg]

Hello!

Is it possible to retrieve the lagrangian which is used in e.g. IPOPT as a symbolic object? I know the solved dual variables can be retrieved, but I need the actual function, since I want to differentiate the lagrangian w.r.t. some of the varaibles in it. 

Is this doable in casadi as is?

Best,

Markus Svedberg

[Joel Andersson]

The way the Lagrangian is defined in CasADi is simply:  f(x) + trans(lam_g)*g(x). We usually do not include the Lagrange multipliers corresponding to simple bounds (bounds on x), since they will just result in an additive contribution to the gradient.

We also handle inequality constraints the same way as equality constraints by combining the multipliers corresponding to upper and lower bounds.

So if you have symbolic expressions corresponding to the decision variable (x), the objective function (f), the constraint function (g), you can construct the gradient of the Lagrangian as follows:

# Multipliers corresponding to bounds on x

lam_x = MX.sym("lam_x", x.sparsity())

# Multipliers corresponding to bounds on g(x)

lam_g = MX.sym("lam_g", g.sparsity())

# Lagrangian formulation

L = f + dot(lam_g, g) + dot(lam_x, x)

# Gradient of the Lagrangian

grad_L = gradient(L, x)

# Lagrangian formulation with lam_x terms excluded (this is how they are normally formulated in CasADi)

L_alt = f + dot(lam_g, g)

# Gradient of the Lagrangian (equivalent to grad_L above)

grad_L_alt = gradient( L_alt, x) + lam_x

You can find some more detail in https://optimization-online.org/2018/05/6642/

Best regards,

Joel

[Markus Svedberg]

Joel,

Thank you very much, I will try this! I actually need to find the lagragian w.r.t. some of the parameters of problem, but I assume a similar procedure holds in that case.

Best,

Markus

[Joel Andersson]

Right, you can differentiate the Lagrangian with respect to some parameters that are not decision variables. That's how CasADi calculates the multipliers corresponding to fixed parameters (lam_p). As explained in that paper. In that case you can just use L(x, lam_g, p) :=  f(x, p) + trans(lam_g)*g(x, p).

Best regards,

Joel

[Joel Andersson]

Or maybe you meant a subset of x. Anyway, that works too with the same approach. Joel

[Markus Svedberg]

Joeal,

No, a subset of p is what I meant. 

I will try what you suggested

 L(x, lam_g, p) :=  f(x, p) + trans(lam_g)*g(x, p). 

Best,

Markus