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Grassland: A Rapid Algebraic Modeling System for Million-variable Optimization

Xihan Li, Xiongwei Han, Zhishuo Zhou, Mingxuan Yuan, Jia Zeng, Jun Wang

TL;DR

Grassland is a rapid AMS that provides an end-to-end solution to tackle emerged new challenges, and integrates a parallelized instantiation scheme for large-scale linear constraints, and a sequential decomposition method that accelerates model solving exponentially with an acceptable loss of optimality.

Abstract

An algebraic modeling system (AMS) is a type of mathematical software for optimization problems, which allows users to define symbolic mathematical models in a specific language, instantiate them with given source of data, and solve them with the aid of external solver engines. With the bursting scale of business models and increasing need for timeliness, traditional AMSs are not sufficient to meet the following industry needs: 1) million-variable models need to be instantiated from raw data very efficiently; 2) Strictly feasible solution of million-variable models need to be delivered in a rapid manner to make up-to-date decisions against highly dynamic environments. Grassland is a rapid AMS that provides an end-to-end solution to tackle these emerged new challenges. It integrates a parallelized instantiation scheme for large-scale linear constraints, and a sequential decomposition method that accelerates model solving exponentially with an acceptable loss of optimality. Extensive benchmarks on both classical models and real enterprise scenario demonstrate 6 ~ 10x speedup of Grassland over state-of-the-art solutions on model instantiation. Our proposed system has been deployed in the large-scale real production planning scenario of Huawei. With the aid of our decomposition method, Grassland successfully accelerated Huawei's million-variable production planning simulation pipeline from hours to 3 ~ 5 minutes, supporting near-real-time production plan decision making against highly dynamic supply-demand environment.

Grassland: A Rapid Algebraic Modeling System for Million-variable Optimization

TL;DR

Grassland is a rapid AMS that provides an end-to-end solution to tackle emerged new challenges, and integrates a parallelized instantiation scheme for large-scale linear constraints, and a sequential decomposition method that accelerates model solving exponentially with an acceptable loss of optimality.

Abstract

An algebraic modeling system (AMS) is a type of mathematical software for optimization problems, which allows users to define symbolic mathematical models in a specific language, instantiate them with given source of data, and solve them with the aid of external solver engines. With the bursting scale of business models and increasing need for timeliness, traditional AMSs are not sufficient to meet the following industry needs: 1) million-variable models need to be instantiated from raw data very efficiently; 2) Strictly feasible solution of million-variable models need to be delivered in a rapid manner to make up-to-date decisions against highly dynamic environments. Grassland is a rapid AMS that provides an end-to-end solution to tackle these emerged new challenges. It integrates a parallelized instantiation scheme for large-scale linear constraints, and a sequential decomposition method that accelerates model solving exponentially with an acceptable loss of optimality. Extensive benchmarks on both classical models and real enterprise scenario demonstrate 6 ~ 10x speedup of Grassland over state-of-the-art solutions on model instantiation. Our proposed system has been deployed in the large-scale real production planning scenario of Huawei. With the aid of our decomposition method, Grassland successfully accelerated Huawei's million-variable production planning simulation pipeline from hours to 3 ~ 5 minutes, supporting near-real-time production plan decision making against highly dynamic supply-demand environment.

Paper Structure

This paper contains 33 sections, 4 theorems, 5 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Efficient Instantiation of Linear Constraints in Optimization
  4. Preliminaries
  5. Problem description and challenges
  6. Problem description
  7. Challenges of model instantiation
  8. A principled scheme of efficient linear constraint instantiation
  9. A new AML with symbolic multidimensional expression
  10. An efficient model instantiation algorithm
  11. Parallelization of the model instantiation algorithm
  12. Sequential Decomposition of Large-scale Optimization
  13. Preliminaries: Sequential Decision Making and Rolling Horizon
  14. Forward Rolling Horizon
  15. Guided Rolling Horizon
  16. ...and 18 more sections

Key Result

Lemma 3.1

For expression tree of $expr_G$, without loss of generality, we assume that every path from a leaf node to the root will go through at least one sum operator.

Figures (7)

  • Figure 1: A toy example of mathematical optimization pipeline for practical business decision making scenarios.
  • Figure 2: The expression tree of expression \ref{['eq:expr_i']}, with global index placeholder $i$ and local index placeholder $j$.
  • Figure 3: An example of data partition.
  • Figure 4: Forward rolling horizon and proposed decomposition methods for large-scale mathematical optimization.
  • Figure 5: The fine-tuning procedure. The grey variables are fixed while the green variables are to be re-optimized. State variables whose value are determined by other variables keep free in the whole sequence.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.1
  • Proposition 3.2