Solving Sparse Mixed-Integer Quadratic Problems: Application to the Unit Commitment Problem with Optimal Power Flow

Mixed-Integer Quadratically Constrained Quadratic Programs (MIQCQP) arise in a variety of applications, particularly in energy, water, and gas systems, where discrete decisions interact with nonconvex quadratic constraints. These problems are computationally challenging due to the combination of combinatorial complexity and nonconvexity, often rendering traditional exact methods ineffective for large-scale instances. We propose a solution framework for sparse MIQCQPs that integrates semidefinite programming relaxations with chordal decomposition techniques to exploit sparsity. By leveraging problem structure, we significantly reduce the size of the semidefinite constraints into smaller, tractable blocks, improving the scalability of the relaxation and the overall branch-and-bound procedure. We evaluate our framework on the Unit Commitment problem with AC Optimal Power Flow constraints, a practically relevant and highly challenging problem that couples discrete generation decisions with nonlinear AC network physics. Computational results on standard IEEE test cases with up to 118 buses show that our approach provides strong bounds and high-quality solutions, while scaling significantly better than state-of-the-art global optimization solvers.