Optimality Conditions for the Minimization of Quadratic 0-1 Problems

Abstract

In this paper, we present some explicit local and global optimality conditions for unconstrained binary quadratic problems. Local optimality conditions are categorized by the neighborhood, including 1-flip, 2-flip, and $k$-flip neighborhoods. After defining an evaluation vector and an evaluation matrix, we propose algorithms that make it possible for us to obtain 1-flip and/or 2-flip local solutions in polynomial time. We provide a key insight into the relationship between the local optimality conditions and the global optimality conditions. A discussion of global geometric optimality conditions is also presented in this paper.