Beating the SDP bound for the floor layout problem: A simple combinatorial idea

For many mixed-integer programming (MIP) problems, high-quality dual bounds can be obtained either through advanced formulation techniques coupled with a state-of-the-art MIP solver, or through semi-definite programming (SDP) relaxation hierarchies. In this paper, we introduce an alternative bounding approach that exploits the ‘combinatorial implosion’ effect by solving portions of the original problem and aggregating this information to obtain a global dual bound. We apply this technique to the one-dimensional and two-dimensional floor layout problems and compare it with the bounds generated by both state-of-the-art MIP solvers and by SDP relaxations. Specifically, we prove that the bounds obtained through the proposed technique are at least as good as those obtained through SDP relaxations, and present computational results that these bounds can be significantly stronger and easier to compute than these alternative strategies, particularly for very difficult problem instances.United States. National Science Foundation. Graduate Research Fellowship Program (Grant 1122374)United States. National Science Foundation. Graduate Research Fellowship Program (Grant CMMI-1351619

Strong mixed-integer formulations for the floor layout problem

The floor layout problem (FLP) tasks a designer with positioning a collection of rectangular boxes on a fixed floor in such a way that minimizes total communication costs between the components. While several mixed integer programming (MIP) formulations for this problem have been developed, it remains extremely challenging from a computational perspective. This work takes a systematic approach to constructing MIP formulations and valid inequalities for the FLP that unifies and recovers all known formulations for it. In addition, the approach yields new formulations that can provide a significant computational advantage and can solve previously unsolved instances. While the construction approach focuses on the FLP, it also exemplifies generic formulation techniques that should prove useful for broader classes of problems.United States. National Science Foundation. Graduate Research Fellowship Program (Grant 1122374)United States. National Science Foundation. Graduate Research Fellowship Program (Grant CMMI-1351619

Neural Network Verification as Piecewise Linear Optimization: Formulations for the Composition of Staircase Functions

We present a technique for neural network verification using mixed-integer programming (MIP) formulations. We derive a \emph{strong formulation} for each neuron in a network using piecewise linear activation functions. Additionally, as in general, these formulations may require an exponential number of inequalities, we also derive a separation procedure that runs in super-linear time in the input dimension. We first introduce and develop our technique on the class of \emph{staircase} functions, which generalizes the ReLU, binarized, and quantized activation functions. We then use results for staircase activation functions to obtain a separation method for general piecewise linear activation functions. Empirically, using our strong formulation and separation technique, we can reduce the computational time in exact verification settings based on MIP and improve the false negative rate for inexact verifiers relying on the relaxation of the MIP formulation