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

TL;DR

This paper introduces an exact reformulation method for logic constraints in nonlinear optimization and control, avoiding mixed-integer programming and improving scalability and efficiency.

Contribution

It provides a novel approach to convert logical constraints into differentiable, binary-variable-free expressions suitable for nonlinear programming.

Findings

Achieves exact reformulation of logical constraints

Demonstrates improved solution efficiency on benchmarks

Preserves the original feasible set through smoothing

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.