ReLU Networks as Surrogate Models in Mixed-Integer Linear Programs
This work addresses efficiency improvements for optimization problems involving neural networks, but it is incremental as it builds on existing MILP formulations with enhanced bound tightening.
The paper tackles the problem of embedding ReLU networks as surrogate models in mixed-integer linear programming (MILP) by developing bound tightening procedures that use both input and output bounds, resulting in considerably reduced solution times and demonstrating the suitability of small-sized networks for this purpose.
We consider the embedding of piecewise-linear deep neural networks (ReLU networks) as surrogate models in mixed-integer linear programming (MILP) problems. A MILP formulation of ReLU networks has recently been applied by many authors to probe for various model properties subject to input bounds. The formulation is obtained by programming each ReLU operator with a binary variable and applying the big-M method. The efficiency of the formulation hinges on the tightness of the bounds defined by the big-M values. When ReLU networks are embedded in a larger optimization problem, the presence of output bounds can be exploited in bound tightening. To this end, we devise and study several bound tightening procedures that consider both input and output bounds. Our numerical results show that bound tightening may reduce solution times considerably, and that small-sized ReLU networks are suitable as surrogate models in mixed-integer linear programs.