Homotopy tracing of the minimum-2-norm solution under rank deficiency

Determine whether a homotopy algorithm can trace, for every parameter value, the unique minimum-2-norm minimizer selected from the polytope of constrained rank-deficient quadratic-program minimizers.

Background

The paper’s main path results assume that the reduced Hessian on each visited free block is positive definite. When this condition fails, the quadratic program may have multiple minimizers, so the solution is no longer inherently a single curve. The authors explain that one can select a unique minimum-2-norm minimizer at each parameter value because the constrained minimizers form a polytope on which both the fit and the 1 norm are constant, but they explicitly leave unresolved whether any homotopy method can trace this selected solution continuously or algorithmically across the path.

References

The selection, at least, is settled by the structure rather than by any algorithm: the constrained minimisers form a polytope on which both the fit and the $\ell_1$ norm are constant, so the minimum-$\ell_2$-norm choice exists and is unique at every $\lambda$. Whether any homotopy traces it is the most interesting question we leave open.

— The Critical Line Algorithm and the Constrained LASSO: One Curve, Two Literatures  (2609.25704 - Schmelzer et al., 22 Sep 2026) in Section 6, Conclusion; see also Remark 2.1, “Rank deficiency, and the limits of the standing assumption”