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I am solving a feasibility (No objective) problem in IPOPT. I got the following output:

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I see that the violation of the constraints are of order 1e-15. What is the meaning of dual infeasibility 1e-07 and the overall NLP error 2e-09? I mean what do they represent if I only have constraints?

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2 Answers 2

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It took some digging but I found the answers in an old email thread. The paper linked there is archived here.

$$E_{\mu}(x,\lambda,z) = \max \Big\{ \frac{\| \nabla f(x) + \nabla c(x)\lambda + z \|_\infty}{s_d}, \|c(x)\|_\infty,\frac{\| \text{Diag}(x)\text{Diag(z)}e-\mu e\|_\infty}{s_c} \Big\}$$ Is the the overall NLP error. Dual infeasibility is calculated by $$\| \nabla f(x) + \nabla c(x)\lambda + z \|_\infty$$

The scalings $s_d$, $s_c$ are explained in the paper on page four. A $s_d$ value of around $50$ would explain your observation, such a value is allowed to occur. For the meaning of these in the constraint satisfaction case just think $\nabla f(x) = 0$.

I haven't confirmed numerically if this always holds, so let me know if you have issues. Then someone would have to dive into the code of IpOpt to find the true answer.

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For a feasibility problem, you can think of the primal objective function as $f(x) =0$. These results indicate that your problem was solved—that is, IPOPT found a feasible point.

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