Global minimization of difference of quadratic and convex functions over box or binary constraints

In this paper, we present necessary as well as sufficient conditions for a given feasible point to be a global minimizer of the difference of quadratic and convex functions subject to bounds on the variables. We show that the necessary conditions become necessary and sufficient for global minimizers in the case of a weighted sum of squares minimization problems. We obtain sufficient conditions for global optimality by first constructing quadratic underestimators and then by characterizing global minimizers of the underestimators. We also derive global optimality conditions for the minimization of the difference of quadratic and convex functions over binary constraints. We discuss several numerical examples to illustrate the significance of the optimality conditions. The authors are grateful to the referees for their helpful comments and valuable suggestions which have contributed to the final preparation of the paper.