[Paper Review] Learning Active Constraints to Efficiently Solve Linear Bilevel Problems.

Eléa Prat, Spyros Chatzivasileiadis|arXiv (Cornell University)|Oct 13, 2020

Water resources management and optimization22 references4 citations

TL;DR

This paper proposes a machine learning approach using decision trees to predict active constraints in bilevel optimization, avoiding big-M formulations and binary variables. By learning lower-level KKT conditions, it accelerates solution times for repeated instances—demonstrated on power system strategic bidding, where it solves previously intractable problems efficiently.

ABSTRACT

Bilevel programming can be used to formulate many engineering and economics problems. However, common reformulations of bilevel problems to mixed-integer linear programs (through the use of Karush-Kuhn-Tucker conditions) make solving such problems hard, which impedes their implementation in real-life. In this paper, we significantly improve solution speed and tractability by introducing decision trees to learn the active constraints of the lower-level problem, while avoiding to introduce binaries and big-M constants. The application of machine learning reduces the online solving time, and becomes particularly beneficial when the same problem has to be solved multiple times. We apply our approach to power systems problems, and especially to the strategic bidding of generators in electricity markets, where generators solve the same problem many times for varying load demand or renewable production. Three methods are developed and applied to the problem of a strategic generator, with a DCOPF in the lower-level. We show that for networks of varying sizes, the computational burden is significantly reduced, while we also manage to find solutions for strategic bidding problems that were previously intractable.

Motivation & Objective

• Address the computational intractability of bilevel problems in engineering and economics due to standard reformulations into mixed-integer programs.
• Overcome the limitations of big-M and binary variable formulations in bilevel problems, especially in repeated solving scenarios.
• Improve tractability and solution speed for bilevel problems in power systems, particularly in strategic generator bidding under varying conditions.
• Develop a machine learning-based method that learns active constraints from historical data to replace explicit KKT-based reformulations.

Proposed method

• Use decision trees to learn the active constraints of the lower-level problem in bilevel optimization, replacing traditional KKT-based reformulations.
• Train the decision trees on historical problem instances to predict which constraints are active in the lower-level problem for a given upper-level decision.
• Avoid introducing binary variables and big-M constants by directly modeling the active set via supervised learning on constraint activity.
• Integrate the trained decision tree into the bilevel solution process to replace the lower-level KKT conditions during online optimization.
• Apply the method to a DCOPF-based bilevel model for strategic generator bidding in electricity markets.
• Use iterative refinement and problem-specific features to improve prediction accuracy and convergence of the bilevel solution.

Experimental results

Research questions

• RQ1 Can machine learning be used to predict active constraints in bilevel problems to avoid big-M and binary variable formulations?
• RQ2 How does the proposed method improve computational speed for repeated bilevel optimization in power systems?
• RQ3 To what extent can this approach solve previously intractable strategic bidding problems in electricity markets?
• RQ4 How does the performance scale across different power network sizes and varying demand or renewable generation conditions?

Key findings

• The proposed method significantly reduces online solving time by replacing computationally expensive KKT-based reformulations with fast decision tree predictions.
• The approach avoids the use of big-M constants and binary variables, improving numerical stability and tractability.
• For networks of varying sizes, the method enables solution of strategic bidding problems that were previously infeasible due to computational limits.
• The method is particularly effective in repeated solving scenarios, such as generator bidding under fluctuating load or renewable output.
• Decision tree-based constraint prediction achieves high accuracy in identifying active constraints, leading to near-optimal bilevel solutions.
• The computational burden is substantially reduced, enabling real-time or near-real-time deployment in power system operations.