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The classical αBB method determines univariate quadratic perturbations that convexify twice continuously differentiable functions. This paper generalizes αBB to additionally consider nondiagonal elements in the perturbation Hessian matrix. These correspond to bilinear terms in the underestimators, where previously all nonlinear terms were separable quadratic terms. An interval extension of Gerschgorin's circle theorem guarantees convexity of the underestimator. It is shown that underestimation parameters which are optimal, in the sense that the maximal underestimation error is minimized, can be obtained by solving a linear optimization model.Theoretical results are presented regarding the instantiation of the nondiagonal underestimator that minimizes the maximum error. Two special cases are analyzed to convey an intuitive understanding of that optimally-selected convexifier. Illustrative examples that convey the practical advantage of these new αBB underestimators are presented.

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We derive and study a reformulation technique for general 0–1 quadratic programs (QP) that uses diagonal as well as nondiagonal perturbation of the objective function. The technique is an extension of the Quadratic Convex Reformulation (QCR) method developed by Billionnet and co-workers, adding nondiagonal perturbations, whereas QCR is in a sense diagonal. In this work a set of redundant reformulation-linearization technique (RLT) inequalities are included in the problem. The redundant inequalities are used to induce nondiagonal perturbations of the objective function that improve the bounding characteristics of the continuous relaxation. The optimal convexification is obtained from the solution of a semidefinite program. We apply the nondiagonal QCR (NDQCR) technique to four different types of problems and compare the bounding properties and solution times with the original QCR method. The proposed method outperforms the original QCR method on all four types of test problems.

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