11 research outputs found

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

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    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

    Modeling Combinatorial Disjunctive Constraints via Junction Trees

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    We introduce techniques to build small ideal mixed-integer programming (MIP) formulations of combinatorial disjunctive constraints (CDCs) via the independent branching scheme. We present a novel pairwise IB-representable class of CDCs, CDCs admitting junction trees, and provide a combinatorial procedure to build MIP formulations for those constraints. Generalized special ordered sets (SOSk\text{SOS} k) can be modeled by CDCs admitting junction trees and we also obtain MIP formulations of SOSk\text{SOS} k. Furthermore, we provide a novel ideal extended formulation of any combinatorial disjunctive constraints with fewer auxiliary binary variables with an application in planar obstacle avoidance
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