Polynomial-time pivot rule for the active-set method

Determine whether there exists a polynomial-time pivot rule for the active-set method applied to strictly concave quadratic objective functions.

Background

The paper formulates the active-set method as a nonlinear generalization of the simplex method. Its running time depends on the pivot rule used to select improving directions and update the active constraint set. Although strictly concave quadratic programs admit efficient subproblem computations and are a central setting for active-set algorithms, the paper states that no polynomial running-time guarantee is known for all instances under a suitable pivot rule.

References

It is an open problem whether there is a polynomial time pivot rule for the active-set method for (strictly) concave, quadratic objective functions.

— An unconditional lower bound for the active-set method on the hypercube  (2502.18019 - Disser et al., 25 Feb 2025) in Section 2, subsection “Simplex and active-set method”

Two problems are left open, and both belong to the shared object rather than to either instance. The first is the length of the path: near-linear in practice, exponential in the worst case \citep{mairal2012}, with no tie-break rule known that keeps the corners exact and carries a guarantee.

— The Critical Line Algorithm and the Constrained LASSO: One Curve, Two Literatures  (2609.25704 - Schmelzer et al., 22 Sep 2026) in Section 6, Conclusion