# Some proximity and sensitivity results in quadratic integer programming

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## Abstract

We show that for any optimal solution\(\bar z\) for a given separable quadratic integer programming problem there exist an optimal solution\(\bar x\) for its continuous relaxation such that\(\parallel \bar z - \bar x\parallel _\infty \leqslant n\Delta (A)\) where*n* is the number of variables and*Δ*(*A*) is the largest absolute subdeterminant of the integer constraint matrix*A.* Also for any feasible solution*z*, which is not optimal for the separable quadratic integer programming problem, there exists a feasible solution\(\bar z\) having greater objective function value and with\(\parallel \bar z - z\parallel _\infty \leqslant n\Delta (A)\). We further prove, under some additional assumptions, that the distance between a pair of optimal solutions to an integer quadratic programming problem with right hand side vectors*b* and*b′*, respectively, depends linearly on ‖*b−b′*‖_{1}. Finally the validity of all the results for nonseparable mixed-integer quadratic programs is established. The proximity results obtained in this paper are extensions of some of the results described in Cook et al. (1986) for linear integer programming.

## Key words

Quadratic integer programming proximity analysis sensitivity analysis## Preview

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## References

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