Some proximity and sensitivity results in quadratic integer programming

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\(\parallel \bar z - \bar x\parallel _\infty \leqslant n\Delta (A)\) wheren is the number of variables andΔ(A) is the largest absolute subdeterminant of the integer constraint matrixA. Also for any feasible solutionz, 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 vectorsb andb′, 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.