[Paper Review] A Simple Effective Heuristic for Embedded Mixed-Integer Quadratic Programming

Reza Takapoui, Nicholas Moehle+2 more|IMT Institutional Repository (IMT School for Advanced Studies Lucca)|Sep 28, 2015

Advanced Optimization Algorithms Research81 references4 citations

TL;DR

This paper proposes a fast, ADMM-based heuristic for solving embedded mixed-integer quadratic programs with nonconvex constraints, enabling rapid convergence to high-quality feasible solutions. The method achieves near-global optimality in seconds on embedded platforms, outperforming branch-and-bound in speed while matching or exceeding relax-and-round in solution quality across vehicle control, power conversion, and signal decoding tasks.

ABSTRACT

In this paper we propose a fast optimization algorithm for approximately minimizing convex quadratic functions over the intersection of affine and separable constraints (i.e., the Cartesian product of possibly nonconvex real sets). This problem class contains many NP-hard problems such as mixed-integer quadratic programming. Our heuristic is based on a variation of the alternating direction method of multipliers (ADMM), an algorithm for solving convex optimization problems. We discuss the favorable computational aspects of our algorithm, which allow it to run quickly even on very modest computational platforms such as embedded processors. We give several examples for which an approximate solution should be found very quickly, such as management of a hybrid-electric vehicle drivetrain and control of switched-mode power converters. Our numerical experiments suggest that our method is very effective in finding a feasible point with small objective value; indeed, we find that in many cases, it finds the global solution.

Motivation & Objective

• To develop a computationally efficient heuristic for solving mixed-integer quadratic programs with nonconvex constraints in embedded systems.
• To address the challenge of non-polynomial worst-case runtime in exact methods like branch-and-bound, which are impractical for real-time embedded applications.
• To enable rapid solution of problem instances where parameters such as q, b, and X change between runs, but P and A remain constant.
• To provide a practical alternative to exact methods that balances solution quality, runtime, and implementation simplicity for real-time control and signal processing.
• To demonstrate effectiveness in embedded applications such as hybrid vehicle control, switched-mode power converters, and MIMO signal decoding.

Proposed method

• The method employs a modified alternating direction method of multipliers (ADMM) to solve a convex relaxation of the original nonconvex problem.
• It incorporates a regularization term to penalize deviation from a least-squares solution, improving convergence and solution quality.
• The algorithm alternates between updating primal variables and dual variables using augmented Lagrangian dual ascent steps.
• The nonconvex constraints are enforced via projection steps after each ADMM update, ensuring feasibility in the final iterate.
• The method is initialized with a single starting point and runs for a fixed number of iterations, making it suitable for embedded deployment.
• The approach is designed to be computationally lightweight, enabling execution on modest embedded processors with strict timing constraints.

Experimental results

Research questions

• RQ1 Can a simple ADMM-based heuristic achieve near-global solutions for mixed-integer quadratic programs with nonconvex constraints in real-time embedded systems?
• RQ2 How does the proposed heuristic compare in solution quality and runtime to relax-and-round and branch-and-bound methods in embedded control and signal decoding tasks?
• RQ3 To what extent can the heuristic find the global solution or a feasible point with small objective value in practical embedded applications?
• RQ4 How does the method perform across different initialization strategies and problem instances with varying parameter sets?
• RQ5 Can the heuristic be effectively applied to problems with constant P and A but changing q, b, and X, such as in model predictive control?

Key findings

• The heuristic found a feasible solution with a small objective value in less than 2 seconds for a switched-mode power converter control problem, while MOSEK required over 4 hours to find the global solution.
• In 95% of 1,000 MIMO signal decoding instances, the ADMM-based heuristic achieved a bit error rate (BER) at least as good as the relax-and-round method.
• The objective value of the ADMM solution was better than or equal to the relax-and-round solution in 95% of cases, as shown in the histogram of objective value differences.
• The average runtime per problem instance, including preprocessing, was only 80 milliseconds, significantly faster than branch-and-bound methods.
• For the hybrid vehicle drivetrain control problem, the heuristic delivered a solution with performance practically indistinguishable from the global optimum.
• The method demonstrated robustness across multiple initializations and problem types, consistently finding high-quality feasible solutions in embedded time constraints.