A Posteriori Second-Order Guarantees for Bolza Problems via Collocation

TL;DR

This paper introduces an a posteriori certification method for Bolza optimal control problems solved via collocation, bridging the gap between discrete solver outputs and continuous second-order optimality guarantees.

Contribution

It develops a framework to reconstruct continuous trajectories from discrete solutions and provides a computable second-order sufficiency certificate based on residuals and curvature bounds.

Findings

Provides a verifiable second-order optimality certificate from collocation solutions.

Enables adaptive mesh refinement guided by residual decomposition.

Extends to path inequalities with transversal switches.

Abstract

Direct collocation for Bolza optimal control yields discrete Karush-Kuhn-Tucker (KKT) points, while practical solvers expose only discrete quantities such as primal-dual iterates, reduced Hessians, and Jacobians. This creates a gap between continuous second-order optimality theory and what can be certified from solver output. We develop an a posteriori certification framework that bridges this gap. Starting from a discrete KKT solution, we reconstruct piecewise polynomial state, control, and costate trajectories, evaluate residuals of the dynamics, boundary, and stationarity conditions, and derive a computable lower bound for the continuous second variation. The bound is expressed as the discrete reduced curvature minus explicit residual-dependent correction terms. A positive bound yields a sufficient certificate for continuous second-order sufficiency and provides quantitative information relevant to local growth and trust-region sizing. The constants entering the certification inequality are conservatively estimable from reconstructed discrete data. The resulting test is operationally verifiable from collocation outputs and naturally supports adaptive mesh refinement through residual decomposition. We also outline an extension to path inequalities with isolated transversal switches.