Papers › A Slightly Lifted Convex Relaxation for Nonconvex Quadratic Programming with Ball Constraints
A Slightly Lifted Convex Relaxation for Nonconvex Quadratic Programming with Ball Constraints
Samuel Burer
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
Globally optimizing a nonconvex quadratic over the intersection of m balls in ℝⁿ is known to be polynomial-time solvable for fixed m. Moreover, when m=1, the standard semidefinite relaxation is exact. When m=2, it has been shown recently that an exact relaxation can be constructed using a disjunctive semidefinite formulation based essentially on two copies of the m=1 case. However, there is no known explicit, tractable, exact convex representation for m ≥3. In this paper, we construct a new, polynomially sized semidefinite relaxation for all m, which does not employ a disjunctive approach. We show that our relaxation is exact for m=2. Then, for m ≥3, we demonstrate empirically that it is fast and strong compared to existing relaxations. The key idea of the relaxation is a simple lifting of the original problem into dimension n+1. Extending this construction: (i) we show that nonconvex quadratic programming over ‖x‖ ≤min{ 1, g + hᵀ x } has an exact semidefinite representation; and (ii) we construct a new relaxation for quadratic programming over the intersection of two ellipsoids, which globally solves all instances of a benchmark collection from the literature.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections