When Descent Is Too Stable: Event-Triggered Hamiltonian Learning to Optimize

Fixed-budget nonconvex optimization can fail not because local descent is unstable, but because it is too stable: after reaching a nearby stationary point, an optimizer may spend the remaining evaluations refining an uninformative local minimum. We formulate this failure mode as a control problem over optimizer dynamics, where the learner must decide when to descend, when to exploit a promising basin, and when stagnation should trigger movement elsewhere. We introduce SHAPE, a structured adaptive port-Hamiltonian task-family optimizer for event-triggered minima hunting under local information. Starting from gradient-descent dynamics, SHAPE lifts optimization to an augmented phase space $(q, p)$, where the primal state $q$ represents the candidate solution, the cotangent variable $p$ carries directional sensitivity, and a controller $u$ provides processed information from current gradient oracle. Within each stage, a learned Hamiltonian vector field induces structured local descent; across stages, a fixed event clock in the implementation updates ports and memory when local equilibria are detected, with stage-dependent horizons treated in the analysis as a direct generalization. This design preserves a passivity-compatible structure while allowing the same trained policy to use clean, stochastic, or estimated gradient inputs. Experiments on fixed-budget nonconvex optimization tasks show that SHAPE improves best-so-far performance compared with fixed-policy optimizers. These results suggest that adaptive Hamiltonian energy shaping provides a principled mechanism for balancing descent, exploration, and budget allocation in difficult optimization landscapes.