Bounds for Binary Integer Programming via Unconstrained Quadratic Pseudo-Boolean Optimization

This research proposes a method for finding bounds in binary integer programming (BIP) and quadratic binary integer programming (QBIP). Our approach utilizes a max-flow technique within the framework of quadratic pseudo-Boolean optimization. We apply Lagrange multipliers to transform these problems into a quadratic pseudo-Boolean format, enabling us to create a corresponding network. We develop sufficient conditions under which a flow can or cannot be established in this network. The upper bound of the original problem is obtained through a feasible flow within the network. Additionally, we provide the optimality conditions for the solutions discussed in this paper. We have derived the closed forms for the various functions and values commonly used in our proposed method, as well as a lower bound for both BIP and QBIP, which can assist in solving these problems. This paper includes examples that illustrate both networks containing flows and those that do not, as well as examples that demonstrate lower and upper bound solutions as identified by our method. Furthermore, since our approach can also be applied to Quadratic Unconstrained Binary Optimization (QUBO), we provide the closed forms for the frequently used functions and values in that context, along with an example showcasing its application to a quadratic unconstrained optimization problem.