Abstract
Zero-one maximization of a quadratic function f(x) is NP-hard. In [4] several equivalent polynomially solvable relaxations are described whose common optimal value, w, yields an upper bound of f*, the zero-one maximum of f(x). One of these relaxations is the maximization of a piecewise linear concave function, R(x), over the full unit hypercube. Using this relaxation we obtain a bound on (w−f*). In the special case where the off-diagonal elements of the Hessian matrix are nonnegative, we show that R(x) coincides with the concave envelope.
Keywords
- Quadratic Zero-One
- Roof duality
- Concave envelope
The partial support of NSF Grant No. ECS85-03212 and AFOSR Grant No. 0271 is gratefully acknowledged.
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© 1989 Springer-Verlag
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Hammer, P.L., Kalantari, B. (1989). A bound on the roof-duality gap. In: Simeone, B. (eds) Combinatorial Optimization. Lecture Notes in Mathematics, vol 1403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083469
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DOI: https://doi.org/10.1007/BFb0083469
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51797-9
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