Abstract
Globally optimizing a nonconvex quadratic over the intersection of m balls in \(\mathbb {R}^n\) 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 \(m \ge 3\). 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 \(m \ge 3\), 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 \(\Vert x\Vert \le \min \{ 1, g + h^T x \}\) 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.
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Notes
These instances are available at https://github.com/A-Eltved/strengthened_sdr.
These instances are available at https://github.com/sburer/soctrust.
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Acknowledgements
The author expresses his sincere thanks to Kurt Anstreicher for an important observation, which ultimately led to the establishment of Theorem 1. Thanks are also extended to the anonymous reviewers and editors, whose suggestions have improved this paper immensely.
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Burer, S. A slightly lifted convex relaxation for nonconvex quadratic programming with ball constraints. Math. Program. (2024). https://doi.org/10.1007/s10107-024-02076-1
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DOI: https://doi.org/10.1007/s10107-024-02076-1