We explore in this paper certain rich geometric properties hidden behind quadratic 0-1 programming. Especially, we derive new lower bounding methods and variable fixation techniques for quadratic 0-1 optimization problems by investigating geometric features of the ellipse contour of a (perturbed) convex quadratic function. These findings further lead to some new optimality conditions for quadratic 0-1 programming. Integrating these novel solution schemes into a proposed solution algorithm of a branch-and-bound type, we obtain promising preliminary computational results.

Abstract. We present in this paper new results on the duality gap between the binary quadratic optimization problem and its Lagrangian dual or semidefinite programming relaxation. We first derive a necessary and sufficient condition for the zero duality gap and discuss its relationship with the polynomial solvability of the primal problem. We then characterize the zeroness of the duality gap by the distance, δ, between {−1, 1} n and certain affine subspace C and show that the duality gap can be reduced by an amount proportional to δ 2 . We finally establish the connection between the computation of δ and cell enumeration of hyperplane arrangement in discrete geometry and further identify two polynomially solvable cases of computing δ.