Inverse optimization asks: given someone's decisions, what objective were they optimizing? The assumption is that enough observations will pin down the answer. A comprehensive framework for parametric convex inverse optimization now proves this assumption wrong. Non-identifiability is the generic case.
Even with normalization constraints, even with multiple observations, the set of parameters compatible with the data is generically multi-dimensional. Not a point, not even a curve — a volume. Regularization doesn't resolve it. It merely selects an element from the ambiguous set, disguising the degeneracy as a unique answer.
The response isn't despair but redirection. Instead of recovering the unknown parameter (which you provably can't), recover the latent optimal solution — what the decision-maker would do next. The set of compatible parameters is large, but the set of implied future decisions can be small. The map from parameters to decisions is many-to-one, and it's the “one” that matters.
The framework, called Inverse Learning, achieves something unusual: a complexity reduction that doesn't depend on how many observations you have. More data doesn't make the parameter set smaller. But it does make the implied decision set tighter. The inferential target was wrong. Once corrected, the problem becomes tractable — not because the ambiguity was resolved, but because it was recognized as irrelevant.