mixed integer linear programming verification

**MILP** (Mixed-Integer Linear Programming) Verification is the **encoding of neural network verification problems as mixed-integer optimization problems** — where ReLU activations are modeled as binary variables and the verification question becomes an optimization feasibility problem. **How MILP Verification Works** - **Linear Layers**: Encoded directly as linear constraints ($y = Wx + b$). - **ReLU**: Modeled with binary variable $z in {0, 1}$: $y leq x - l(1-z)$, $y geq x$, $y leq uz$, $y geq 0$. - **Objective**: Maximize (or check feasibility of) the target property violation. - **Solver**: Commercial solvers (Gurobi, CPLEX) solve the MILP with branch-and-bound. **Why It Matters** - **Exact**: MILP provides exact verification — no approximation, no false positives. - **Flexible**: Can encode complex properties (multi-class robustness, output constraints). - **State-of-Art**: Combined with bound tightening (CROWN bounds), MILP-based tools win verification competitions. **MILP Verification** is **optimization-based proof** — encoding neural network properties as integer programs for exact formal verification.