Solving Mixed-Integer Quadratic Programs via Nonnegative Least Squares

This paper proposes a new algorithm for solving Mixed-Integer Quadratic Programming (MIQP) problems. The algorithm is particularly tailored to solving small-scale MIQPs such as those that arise in embedded hybrid Model Predictive Control (MPC) applications. The approach combines branch and bound (B&B) with nonnegative least squares (NNLS), that are used to solve Quadratic Programming (QP) relaxations. The QP algorithm extends a method recently proposed by the author for solving strictly convex QP's, by (i) handling equality and bilateral inequality constraints, (ii) warm starting, and (iii) exploiting easy-to-compute lower bounds on the optimal cost to reduce the number of QP iterations required to solve the relaxed problems. The proposed MIQP algorithm has a speed of execution that is comparable to state- of-the-art commercial MIQP solvers and is relatively simple to code, as it requires only basic arithmetic operations to solve least-square problems.