Exact and Heuristic Methods with Warm-start for Embedded Mixed-Integer Quadratic Programming Based on Accelerated Dual Gradient Projection

Abstract: Small-scale Mixed-Integer Quadratic Programming (MIQP) problems often arise in embedded control and estimation applications. Driven by the need for algorithmic simplicity to target computing platforms with limited memory and computing resources, this paper proposes a few approaches to solving MIQPs, either to optimality or suboptimally. We specialize an existing Accelerated Dual Gradient Projection (GPAD) algorithm to effectively solve the Quadratic Programming (QP) relaxation that arise during Branch and Bound (B&B) and propose a generic framework to warm-start the binary variables which reduces the number of QP relaxations. Moreover, in order to find an integer feasible combination of the binary variables upfront, two heuristic approaches are presented: ($i$) without using B&B, and ($ii$) using B&B with a significantly reduced number of QP relaxations. Both heuristic approaches return an integer feasible solution that may be suboptimal but involve a much reduced computation effort. Such a feasible solution can be either implemented directly or used to set an initial upper bound on the optimal cost in B&B. Through different hybrid control and estimation examples involving binary decision variables, we show that the performance of the proposed methods, although very simple to code, is comparable to that of state-of-the-art MIQP solvers.