Smooth Logic Constraints in Nonlinear Optimization and Optimal Control Problems

TL;DR

This paper introduces a novel method for embedding complex logical constraints into nonlinear optimization problems by reformulating and smoothing them, demonstrated on quadrotor control tasks with improved efficiency and convergence.

Contribution

It presents a new approach to incorporate logic constraints directly into continuous optimization, avoiding binary variables and specialized solvers.

Findings

Faster computational performance compared to existing methods

Improved convergence to feasible solutions

Effective handling of complex logic in control tasks

Abstract

In some optimal control problems, complex relationships between states and inputs cannot be easily represented using continuous constraints, necessitating the use of discrete logic instead. This paper presents a method for incorporating such logic constraints directly within continuous optimization frameworks, eliminating the need for binary variables or specialized solvers. Our approach reformulates arbitrary logic constraints under minimal assumptions as max-min constraints, which are then smoothed by introducing auxiliary variables into the optimization problem. When these reformulated constraints are satisfied, they guarantee that the original logical conditions hold, ensuring correctness in the optimization process. We demonstrate the effectiveness of this method on two planar quadrotor control tasks with complex logic constraints. Compared to existing techniques for encoding logic in continuous optimization, our approach achieves faster computational performance and improved convergence to feasible solutions.