Abstract
This paper develops new semidefinite programming (SDP) relaxation techniques for two classes of mixed binary quadratically constrained quadratic programs and analyzes their approximation performance. The first class of problems finds two minimum norm vectors in N-dimensional real or complex Euclidean space, such that M out of 2M concave quadratic constraints are satisfied. By employing a special randomized rounding procedure, we show that the ratio between the norm of the optimal solution of this model and its SDP relaxation is upper bounded by \(\frac{54M^2}{\uppi }\) in the real case and by \(\frac{24M}{\sqrt{\uppi }}\) in the complex case. The second class of problems finds a series of minimum norm vectors subject to a set of quadratic constraints and cardinality constraints with both binary and continuous variables. We show that in this case the approximation ratio is also bounded and independent of problem dimension for both the real and the complex cases.
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Notes
There is another way to do the relaxation, i.e., the SDP relaxation for both the discrete and the continuous constraints. However, it can be similar to that in [19] to prove that the two SDP relaxation problems are equivalent. Due to its smaller problem size, the SDP relaxation problem (SDP1) is preferred.
The probability that no such \({{\xi }}^{(1)}\) or \({{\xi }}^{(2)}\) is generated after N independent trials is at most \((1-0.075\,8\cdots )^N\). When \(N=100\), the probability equals \(0.000\,375\cdots \). Thus, such \({{\xi }}^{(1)}\) or \({{\xi }}^{(2)}\) requires relatively few trials to generate, see also in [2].
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The first author’s research was supported by the National Natural Science Foundation of China (No. 11101261).
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Xu, Z., Hong, MY. Semidefinite Relaxation for Two Mixed Binary Quadratically Constrained Quadratic Programs: Algorithms and Approximation Bounds. J. Oper. Res. Soc. China 4, 205–221 (2016). https://doi.org/10.1007/s40305-015-0082-2
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DOI: https://doi.org/10.1007/s40305-015-0082-2
Keywords
- Nonconvex quadratically constrained quadratic programming
- Semidefinite program relaxation
- Approximation bound
- NP-hard