Second-order optimality conditions for stationary points

Investigate second-order conditions for the nonconvex interacting kinetic Schrödinger bridge optimization problems to characterize the nature of the stationary points produced by the derived first-order necessary conditions.

Background

The paper formulates nonconvex minimum-energy steering problems for mean-field inertial swarms with Cucker–Smale or Morse interactions. Because the interaction depends on the evolving swarm density, the derived forward–backward systems provide only first-order necessary conditions, and the numerical method computes stationary points rather than establishing global optimality. The authors therefore identify the characterization of these stationary points through second-order conditions as a future research direction.

References

The optimization problems considered are nonconvex since the interaction depends on the evolving swarm density; thus, the systems derived here provide first-order necessary conditions for optimality, and the computed solutions are stationary points rather than global minima. A future research direction is therefore to investigate second-order conditions to characterize the nature of the stationary points.

— Schrödinger Bridges over Kinetic Swarming Models  (2608.25281 - Eldesoukey et al., 26 Aug 2026) in Section Conclusions