A slightly lifted convex relaxation for nonconvex quadratic programming with ball constraints
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 disjunctiv...
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| Vydané v: | Mathematical programming Ročník 211; číslo 1-2; s. 157 - 179 |
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| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Heidelberg
Springer Nature B.V
01.05.2025
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| Predmet: | |
| ISSN: | 0025-5610, 1436-4646 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | 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+hTx} 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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| Bibliografia: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0025-5610 1436-4646 |
| DOI: | 10.1007/s10107-024-02076-1 |