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Hybrid optimal control with mixed-integer Lagrangian methods

Viktoriya Nikitina, Alberto De Marchi, Matthias Gerdts

TL;DR

The paper addresses hybrid optimal control problems with mixed-integer and nonsmooth dynamics, where standard relaxation or decomposition methods may be inadequate. It adopts a discretize-then-optimize MINLP approach and solves it with a safeguarded augmented Lagrangian, using inner successive mixed-integer linearizations with trust regions. Two challenging benchmarks—a car with hysteretic turbo and a Lotka-Volterra fishing problem with total variation—demonstrate the method's ability to produce good local solutions with favorable runtimes and to control chattering. The results show that the proposed framework provides a practical, scalable alternative for hybrid OCPs without relaxing integrality, suitable for diverse applications.

Abstract

Models involving hybrid systems are versatile in their application but difficult to optimize efficiently due to their combinatorial nature. This work presents a method to cope with hybrid optimal control problems which, in contrast to decomposition techniques, does not require relaxing the integrality constraints. Based on the discretize-then-optimize approach, our scheme addresses mixed-integer nonlinear problems under mild assumptions. The proposed numerical algorithm builds upon the augmented Lagrangian framework, whose subproblems are handled using successive mixed-integer linearizations with trust regions. We validate the performance of the numerical routine with extensive investigations using hybrid optimal control problems from different fields of application. Promising preliminary results are presented for a motion planning task with hysteresis and a Lotka-Volterra fishing problem with total variation.

Hybrid optimal control with mixed-integer Lagrangian methods

TL;DR

The paper addresses hybrid optimal control problems with mixed-integer and nonsmooth dynamics, where standard relaxation or decomposition methods may be inadequate. It adopts a discretize-then-optimize MINLP approach and solves it with a safeguarded augmented Lagrangian, using inner successive mixed-integer linearizations with trust regions. Two challenging benchmarks—a car with hysteretic turbo and a Lotka-Volterra fishing problem with total variation—demonstrate the method's ability to produce good local solutions with favorable runtimes and to control chattering. The results show that the proposed framework provides a practical, scalable alternative for hybrid OCPs without relaxing integrality, suitable for diverse applications.

Abstract

Models involving hybrid systems are versatile in their application but difficult to optimize efficiently due to their combinatorial nature. This work presents a method to cope with hybrid optimal control problems which, in contrast to decomposition techniques, does not require relaxing the integrality constraints. Based on the discretize-then-optimize approach, our scheme addresses mixed-integer nonlinear problems under mild assumptions. The proposed numerical algorithm builds upon the augmented Lagrangian framework, whose subproblems are handled using successive mixed-integer linearizations with trust regions. We validate the performance of the numerical routine with extensive investigations using hybrid optimal control problems from different fields of application. Promising preliminary results are presented for a motion planning task with hysteresis and a Lotka-Volterra fishing problem with total variation.
Paper Structure (9 sections, 20 equations, 4 figures, 1 table)

This paper contains 9 sections, 20 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Optimality Conditions
  3. Mixed-integer polyhedral constraints
  4. Nonlinear constraints
  5. Algorithmic Framework
  6. Numerical Results
  7. Car with hysteretic turbo charger and drag
  8. Lotka-Volterra fishing with total variation
  9. Conclusions

Figures (4)

  • Figure 1: Solutions for the turbo car problem with different values of drag coefficient $c_{\rm d}\in\{10^{-3},10^{-2}\}$. Comparison of \ref{['alg:ALM']} (ALM), starting from an all-zero initial guess, and dynamic programming (DP).
  • Figure 2: Runtimes for the turbo car problem with drag $c_{\rm d}=10^{-3}$ for different number $N$ of discretization intervals. Solutions obtained with ALM and subsequent fixed-integer NLP refinement.
  • Figure 3: Solutions for the fishing problem without $\mathop{\mathrm{TV}}\nolimits$, starting CIA and ALM from the NLP relaxation.
  • Figure 4: Solutions for the fishing problem with limited TV, starting ALM from the CIA solution.

Theorems & Definitions (5)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 3.1