Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning to Solve Constrained Bilevel Control Co-Design Problems

James Kotary, Himanshu Sharma, Ethan King, Draguna Vrabie, Ferdinando Fioretto, Jan Drgona

TL;DR

The paper tackles the bottleneck of solving parametric bilevel optimization problems with coupling constraints by introducing a learning-to-optimize framework that differentiates through both lower-level solvers and constraint projections. It amalgamates a neural predictor with a differentiable coupling-constraint correction routine to produce feasible upper-level solutions across problem distributions, trained via empirical risk minimization. The approach is validated on small bilevel quadratic programs and challenging control-co-design problems (nonlinear two-tank and HVAC systems), demonstrating near-optimal solutions with substantial speedups over traditional metaheuristics and exact solvers. These results indicate strong potential for real-time design and control applications where bilevel structures arise, while also outlining avenues for handling integers and nondifferentiable lower-level objectives in future work.

Abstract

Learning to Optimize (L2O) is a subfield of machine learning (ML) in which ML models are trained to solve parametric optimization problems. The general goal is to learn a fast approximator of solutions to constrained optimization problems, as a function of their defining parameters. Prior L2O methods focus almost entirely on single-level programs, in contrast to the bilevel programs, whose constraints are themselves expressed in terms of optimization subproblems. Bilevel programs have numerous important use cases but are notoriously difficult to solve, particularly under stringent time demands. This paper proposes a framework for learning to solve a broad class of challenging bilevel optimization problems, by leveraging modern techniques for differentiation through optimization problems. The framework is illustrated on an array of synthetic bilevel programs, as well as challenging control system co-design problems, showing how neural networks can be trained as efficient approximators of parametric bilevel optimization.

Learning to Solve Constrained Bilevel Control Co-Design Problems

TL;DR

The paper tackles the bottleneck of solving parametric bilevel optimization problems with coupling constraints by introducing a learning-to-optimize framework that differentiates through both lower-level solvers and constraint projections. It amalgamates a neural predictor with a differentiable coupling-constraint correction routine to produce feasible upper-level solutions across problem distributions, trained via empirical risk minimization. The approach is validated on small bilevel quadratic programs and challenging control-co-design problems (nonlinear two-tank and HVAC systems), demonstrating near-optimal solutions with substantial speedups over traditional metaheuristics and exact solvers. These results indicate strong potential for real-time design and control applications where bilevel structures arise, while also outlining avenues for handling integers and nondifferentiable lower-level objectives in future work.

Abstract

Learning to Optimize (L2O) is a subfield of machine learning (ML) in which ML models are trained to solve parametric optimization problems. The general goal is to learn a fast approximator of solutions to constrained optimization problems, as a function of their defining parameters. Prior L2O methods focus almost entirely on single-level programs, in contrast to the bilevel programs, whose constraints are themselves expressed in terms of optimization subproblems. Bilevel programs have numerous important use cases but are notoriously difficult to solve, particularly under stringent time demands. This paper proposes a framework for learning to solve a broad class of challenging bilevel optimization problems, by leveraging modern techniques for differentiation through optimization problems. The framework is illustrated on an array of synthetic bilevel programs, as well as challenging control system co-design problems, showing how neural networks can be trained as efficient approximators of parametric bilevel optimization.

Paper Structure

This paper contains 23 sections, 25 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Setting
  4. Learning to Solve Bilevel Optimization
  5. Satisfying Constraints with Differentiable Optimization Modules
  6. End-to-End Trainable Architecture
  7. Satisfying Lower-Level Constraints
  8. Experiments
  9. Preliminary Problem: Learning Bilevel Quadratic Programming
  10. Learning Optimal Control Co-Design
  11. Nonconvex Bilevel Optimization: Control Co-design of a Nonlinear System
  12. Control Co-Design of a Building HVAC System
  13. Conclusion and Limitations
  14. Experimental Details: Particle Swarm Optimization
  15. Additional Results: Particle Swarm Optimization
  16. ...and 8 more sections

Figures (6)

  • Figure 1: Illustrating the end-to-end trainable model with algorithm line and equation numbers.
  • Figure 2: Optimality gap and coupling violation, on different-sized BQP problems from left to right: $3 \times 2$, $6 \times 4$, $9 \times 6$. Shows mean and standard deviation over the test set, at each training epoch.
  • Figure 3: Test set metrics, learning control co-design of the two-tank (left) / HVAC (right) systems.
  • Figure 4: Best Objective Value per PSO Iteration on Two-Tank System Co-Design
  • Figure 5: Best Objective Value per PSO Iteration on HVAC System Co-Design
  • ...and 1 more figures