Online Mixed-Integer Optimization in Milliseconds

Dimitris Bertsimas, Bartolomeo Stellato

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


We propose a method to approximate the solution of online mixed-integer optimization (MIO) problems at very high speed using machine learning. By exploiting the repetitive nature of online optimization, we can greatly speed up the solution time. Our approach encodes the optimal solution into a small amount of information denoted as strategy using the voice of optimization framework. In this way, the core part of the optimization routine becomes a multiclass classification problem that can be solved very quickly. In this work, we extend that framework to real-time and high-speed applications focusing on parametric mixed-integer quadratic optimization. We propose an extremely fast online optimization method consisting of a feedforward neural network evaluation and a linear system solution where the matrix has already been factorized. Therefore, this online approach does not require any solver or iterative algorithm. We show the speed of the proposed method both in terms of total computations required and measured execution time. We estimate the number of floating point operations required to completely recover the optimal solution as a function of the problem dimensions. Compared with state-of-the-art MIO routines, the online running time of our method is very predictable and can be lower than a single matrix factorization time. We benchmark our method against the state-of-the-art solver Gurobi obtaining up to two to three orders of magnitude speedups on examples from fuel cell energy management, sparse portfolio optimization, and motion planning with obstacle avoidance.

Original languageEnglish (US)
Pages (from-to)2229-2248
Number of pages20
JournalINFORMS Journal on Computing
Issue number4
StatePublished - Jul 2022

All Science Journal Classification (ASJC) codes

  • Software
  • Information Systems
  • Computer Science Applications
  • Management Science and Operations Research


  • analysis of algorithms: computational complexity
  • artificial intelligence
  • computational methods
  • heuristic
  • mixed-integer optimization


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