Strongly polynomial algorithm for convex quadratic programming (minimization)

Determine whether there exists a strongly polynomial-time algorithm for convex quadratic programming, i.e., minimizing a convex quadratic objective subject to linear inequality constraints, beyond the known weakly polynomial ellipsoid and interior-point methods.

Background

The authors emphasize that while weakly polynomial algorithms exist for convex quadratic programming (via ellipsoid and interior-point methods), it is unknown whether a strongly polynomial algorithm exists.

They view advances on active-set methods and quadratic objectives as promising steps toward resolving this prominent open problem.

References

Furthermore, our result represents significant progress towards concave quadratic (convex quadratic for minimization) objectives, where weakly polynomial algorithms exist and the existence of a strongly polynomial algorithm is a prominent open problem.

— An unconditional lower bound for the active-set method in convex quadratic maximization  (2507.16648 - Bach et al., 22 Jul 2025) in Section 1 (Introduction), Our results subparagraph

We believe that our techniques can be applied to prove that the algorithm of is in fact strongly polynomial, once we remove the normalization $q_{ii}=1$.

— Treewidth and the complexity of box-constrained quadratic programs  (2609.35595 - Pia et al., 28 Sep 2026) in Section 2, subsection Relation to the literature , item (iii) Model of computation