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Exact Continuous Reformulations of Logic Constraints in Nonlinear Optimization and Optimal Control Problems

Jad Wehbeh, Eric C. Kerrigan

TL;DR

The paper tackles the challenge of integrating discrete logic into nonlinear optimization and optimal control without relying on binary variables. It introduces an exact reformulation that transforms logic constraints into binary-free, differentiable expressions by converting to CNF, recasting as max-min constraints, and applying a smoothing step. This results in an NLP-ready formulation with $r$-differentiability and linear growth in constraint count, demonstrated on a quadrotor with obstacle logic and a two-tank system with temporal logic, where it outperforms traditional binary-based methods in success rate and speed. The approach facilitates direct use of standard nonlinear solvers for logic-constrained problems and highlights the trade-offs linked to the choice of logical representation and problem size, suggesting broad applicability to nonlinear OCPs with moderate logical complexity.

Abstract

Many nonlinear optimal control and optimization problems involve constraints that combine continuous dynamics with discrete logic conditions. Standard approaches typically rely on mixed-integer programming, which introduces scalability challenges and requires specialized solvers. This paper presents an exact reformulation of broad classes of logical constraints as binary-variable-free expressions whose differentiability properties coincide with those of the underlying predicates, enabling their direct integration into nonlinear programming models. Our approach rewrites arbitrary logical propositions into conjunctive normal form, converts them into equivalent max--min constraints, and applies a smoothing procedure that preserves the exact feasible set. The method is evaluated on two benchmark problems, a quadrotor trajectory optimization with obstacle avoidance and a hybrid two-tank system with temporal logic constraints, and is shown to obtain optimal solutions more consistently and efficiently than existing binary variable elimination techniques.

Exact Continuous Reformulations of Logic Constraints in Nonlinear Optimization and Optimal Control Problems

TL;DR

The paper tackles the challenge of integrating discrete logic into nonlinear optimization and optimal control without relying on binary variables. It introduces an exact reformulation that transforms logic constraints into binary-free, differentiable expressions by converting to CNF, recasting as max-min constraints, and applying a smoothing step. This results in an NLP-ready formulation with -differentiability and linear growth in constraint count, demonstrated on a quadrotor with obstacle logic and a two-tank system with temporal logic, where it outperforms traditional binary-based methods in success rate and speed. The approach facilitates direct use of standard nonlinear solvers for logic-constrained problems and highlights the trade-offs linked to the choice of logical representation and problem size, suggesting broad applicability to nonlinear OCPs with moderate logical complexity.

Abstract

Many nonlinear optimal control and optimization problems involve constraints that combine continuous dynamics with discrete logic conditions. Standard approaches typically rely on mixed-integer programming, which introduces scalability challenges and requires specialized solvers. This paper presents an exact reformulation of broad classes of logical constraints as binary-variable-free expressions whose differentiability properties coincide with those of the underlying predicates, enabling their direct integration into nonlinear programming models. Our approach rewrites arbitrary logical propositions into conjunctive normal form, converts them into equivalent max--min constraints, and applies a smoothing procedure that preserves the exact feasible set. The method is evaluated on two benchmark problems, a quadrotor trajectory optimization with obstacle avoidance and a hybrid two-tank system with temporal logic constraints, and is shown to obtain optimal solutions more consistently and efficiently than existing binary variable elimination techniques.
Paper Structure (18 sections, 4 theorems, 25 equations, 5 figures, 2 tables)

This paper contains 18 sections, 4 theorems, 25 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. State of the Art
  3. Contributions
  4. Problem Statement
  5. Continuous Constraint Reformulation
  6. Logic Simplification
  7. Max-Min Reformulation
  8. Constraint Smoothing
  9. Binary-Free Optimal Control Problem Formulation
  10. Common Logical Operators
  11. Implication Operator
  12. Equivalence Operator
  13. Until Operator
  14. Release Operator
  15. Numerical Examples
  16. ...and 3 more sections

Key Result

Lemma 1

Any problem in $\mathcal{L}$ can be rewritten using only inequality propositions, $\land$, and $\lor$ operators.

Figures (5)

  • Figure 1: Obstacle (red), logic trigger (green), target (black), and solution trajectory (blue) for Problem 1.
  • Figure 2: Illustration of the quadrotor's horizontal position ($r$), altitude ($s$), and tilt angle ($\psi$) taken from wehbeh2024robust.
  • Figure 3: Two tank problem setup showing measurements (in black) and water flow (in blue).
  • Figure 4: Optimal Trajectory for Case 1.
  • Figure 5: Optimal Trajectory for Case 2 with the condition on $h^2$ (green) that releases the constraint on $h^1$ (red).

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Lemma 3