Meaning of each column solver output in Gekko

Question

I am curious what each solver output column from IPOPT solver suggests. Is there any material that explains this?

Below is solver output from IPOPT solver. And I'd like to know what the inf_pr, inf_du, lg(mu), ||d||, etc. terms mean.

Answer 1

Below is a description of each column from the IPOPT documentation.

• iter: The current iteration count. This includes regular iterations and iterations during the restoration phase. If the algorithm is in the restoration phase, the letter "r" will be appended to the iteration number.
• objective: The unscaled objective value at the current point. During the restoration phase, this value remains the unscaled objective value for the original problem.
• inf_pr: The unscaled constraint violation at the current point. This quantity is the infinity-norm (max) of the (unscaled) constraints ( gL≤g(x)≤gU in (NLP)). During the restoration phase, this value remains the constraint violation of the original problem at the current point. The option inf_pr_output can be used to switch to the printing of a different quantity.
• inf_du: The scaled dual infeasibility at the current point. This quantity measure the infinity-norm (max) of the internal dual infeasibility, Eq. (4a) in the implementation paper [11], including inequality constraints reformulated using slack variables and problem scaling. During the restoration phase, this is the value of the dual infeasibility for the restoration phase problem.
• lg(mu): log10 of the value of the barrier parameter μ.
• ||d||: The infinity norm (max) of the primal step (for the original variables x and the internal slack variables s). During the restoration phase, this value includes the values of additional variables, p and n (see Eq. (30) in [11]).
• lg(rg): log10 of the value of the regularization term for the Hessian of the Lagrangian in the augmented system ( δw in Eq. (26) and Section 3.1 in [11]). A dash ("-") indicates that no regularization was done.
• alpha_du: The stepsize for the dual variables ( αzk in Eq. (14c) in [11]).
• alpha_pr: The stepsize for the primal variables ( αk in Eq. (14a) in [11]). The number is usually followed by a character for additional diagnostic information regarding the step acceptance criterion:

Tag Description
f   f-type iteration in the filter method w/o second order correction
F   f-type iteration in the filter method w/ second order correction
h   h-type iteration in the filter method w/o second order correction
H   h-type iteration in the filter method w/ second order correction
k   penalty value unchanged in merit function method w/o second order correction
K   penalty value unchanged in merit function method w/ second order correction
n   penalty value updated in merit function method w/o second order correction
N   penalty value updated in merit function method w/ second order correction
R   Restoration phase just started
w   in watchdog procedure
s   step accepted in soft restoration phase
t/T tiny step accepted without line search
r   some previous iterate restored

• ls: The number of backtracking line search steps (does not include second-order correction steps).

There is additional information on interior point methods with code examples on my course website for engineering design optimization and in Section 8.4 of the Design Optimization textbook.