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Small-scale Mixed-Integer Quadratic Programming (MIQP) problems often arisein embedded control and estimation applications. Driven by the need foralgorithmic simplicity to target computing platforms with limited memory andcomputing resources, this paper proposes a few approaches to solving MIQPs,either to optimality or suboptimally. We specialize an existing AcceleratedDual Gradient Projection (GPAD) algorithm to effectively solve the QuadraticProgramming (QP) relaxation that arise during Branch and Bound (B&B) andpropose a generic framework to warm-start the binary variables which reducesthe number of QP relaxations. Moreover, in order to find an integer feasiblecombination of the binary variables upfront, two heuristic approaches arepresented: ($i$) without using B&B, and ($ii$) using B&B with a significantlyreduced number of QP relaxations. Both heuristic approaches return an integerfeasible solution that may be suboptimal but involve a much reduced computationeffort. Such a feasible solution can be either implemented directly or used toset an initial upper bound on the optimal cost in B&B. Through different hybridcontrol and estimation examples involving binary decision variables, we showthat the performance of the proposed methods, although very simple to code, iscomparable to that of state-of-the-art MIQP solvers.

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