A Two-Level Approach to Large Mixed-Integer Programs with Application to Cogeneration in Energy-Efficient Buildings

TL;DR

This paper introduces a two-level coarsening approach for large-scale mixed-integer linear programs, enabling efficient approximate solutions for complex energy cogeneration problems in buildings.

Contribution

The paper presents a novel two-level coarsening method that guarantees feasible solutions and converges to the optimal bound for large MILPs, outperforming existing solvers.

Findings

Effective in solving large MILPs with over 1 million binary variables

Provides good approximate solutions at significantly reduced computational time

Scales beyond the capacity of current commercial solvers

Abstract

We study a two-stage mixed-integer linear program (MILP) with more than 1 million binary variables in the second stage. We develop a two-level approach by constructing a semi-coarse model (coarsened with respect to variables) and a coarse model (coarsened with respect to both variables and constraints). We coarsen binary variables by selecting a small number of pre-specified daily on/off profiles. We aggregate constraints by partitioning them into groups and summing over each group. With an appropriate choice of coarsened profiles, the semi-coarse model is guaranteed to find a feasible solution of the original problem and hence provides an upper bound on the optimal solution. We show that solving a sequence of coarse models converges to the same upper bound with proven finite steps. This is achieved by adding violated constraints to coarse models until all constraints in the semi-coarse model are satisfied. We demonstrate the effectiveness of our approach in cogeneration for buildings. The coarsened models allow us to obtain good approximate solutions at a fraction of the time required by solving the original problem. Extensive numerical experiments show that the two-level approach scales to large problems that are beyond the capacity of state-of-the-art commercial MILP solvers.