Strong Duality and Dual Ascent Approach to Continuous-Time Chance-Constrained Stochastic Optimal Control

2025-11-19

AI Summary

This paper addresses the stochastic optimal control (SOC) problem with probabilistic (chance) constraints in continuous time and continuous state space—specifically, enforcing strict upper bounds on the probability of state constraint violation. We propose a novel method that avoids conservative approximations: by leveraging the first exit time, chance constraints are reformulated as expectations of indicator functions; these are then incorporated into the cost functional via Lagrangian duality, and strong duality is established to link the dualized problem to the Hamilton–Jacobi–Bellman equation. To our knowledge, this is the first rigorous proof of strong duality for chance-constrained SOC in continuous time. Building upon this, we develop a path-integral-based solution framework using dual ascent, enabling efficient gradient optimization via open-loop trajectory sampling. Evaluated on mobile robot navigation tasks, our approach achieves accuracy comparable to finite-difference methods while demonstrating superior scalability and computational efficiency.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Large-scale security measurements

Abstract

The paper addresses a continuous-time continuous-space chance-constrained stochastic optimal control (SOC) problem where the probability of failure to satisfy given state constraints is explicitly bounded. We leverage the notion of exit time from continuous-time stochastic calculus to formulate a chance-constrained SOC problem. Without any conservative approximation, the chance constraint is transformed into an expectation of an indicator function which can be incorporated into the cost function by considering a dual formulation. We then express the dual function in terms of the solution to a Hamilton-Jacobi-Bellman partial differential equation parameterized by the dual variable. Under a certain assumption on the system dynamics and cost function, it is shown that a strong duality holds between the primal chance-constrained problem and its dual. The Path integral approach is utilized to numerically solve the dual problem via gradient ascent using open-loop samples of system trajectories. We present simulation studies on chance-constrained motion planning for spatial navigation of mobile robots and the solution of the path integral approach is compared with that of the finite difference method.

Problem

• Addresses chance-constrained stochastic optimal control in continuous time
• Transforms chance constraints into dual formulation without approximations
• Solves dual problem via path integral gradient ascent method

Innovation

• Dual formulation transforms chance constraints into expectations
• Path integral approach solves dual problem via gradient ascent
• Strong duality holds between primal and dual problems