Featurized Occupation Measures for Structured Global Search in Numerical Optimal Control

Qi Wei; Jianfeng Tao; Haoyang Tan; Hongyu Nie

math.OCMar 17, 2026

Featurized Occupation Measures for Structured Global Search in Numerical Optimal Control

Authors: Qi Wei, Jianfeng Tao, Haoyang Tan, Hongyu Nie

Organizations: School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Abstract

Numerical optimal control is split between globally structured but dimensionally intractable Hamilton--Jacobi--Bellman (HJB) methods and scalable but local trajectory optimization. We introduce Featurized Occupation Measures (FOM), a finite-dimensional primal--dual interface where HJB subsolutions guide primal search and primal occupation data can inform certificate refinement. Explicit and implicit realizations, based on finite Liouville tests and rollout-sampled residuals, are proved asymptotically consistent with the exact occupation-measure LP. FOM also identifies when HJB-type certificates avoid full state-space representation. For factor graphs induced by compatible passivity-based interconnections, blockwise HJB inequalities yield block-additive OM-dual certificates, including blockwise approximate variants. The curse of dimensionality is then shifted from state dimension to interconnection topology. Approximate certificates remain reusable under time shifts and bounded model perturbations, with explicit degradation bounds. A static obstacle-avoidance benchmark illustrates how tighter certificates improve sample-based global search.

Figures & tables

Panel of Figure 2 — Fig. 2 : Certificate $\min_{\phi}v_{\psi}(p_{x},p_{y},\phi)$ at $K=0$ , $50$ , and $300$ , overlaid with greedy-policy rollouts (red). Additional sweeps propagate obstacle information outward from the goal. Even at $K=300$ the certificate remains coarse; it supplies global lower-bound structure, and the primal update refines the trajectories.

Panel of Figure 3 — Fig. 3 : Optimization behavior on the benchmark problem. In (a), thick curves denote the same mean exact-penalty cost over $50$ different random seeds for each FOM variant and over $50$ random initializations for AL-DDP; shaded regions show the corresponding minimum–maximum envelopes. In (b), tighter certificates favor the lower-cost route.

Method	Mean	SD	Min	Max
AL-DDP	8609.0	465.7	7907.8	9945.3
primal-only FOM	8503.4	296.3	7934.7	8664.2
cert-guided FOM ( $K{=}0$ )	8416.0	343.7	7930.1	8677.9
cert-guided FOM ( $K{=}50$ )	8163.9	279.7	7910.2	8654.8
cert-guided FOM ( $K{=}300$ )	7922.8	22.0	7909.6	7970.4

TABLE I : Exact-penalty cost at iteration

100

HJB Equations

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Jul 26, 2026J/K move · Enter open · S save

When Rates Are Geometric: Rate-Certificate Transfer for Contact Splittings in Optimization

George A Kevrekidis

Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218, USA

Optimization Convergence Analysis Hamiltonian Dynamics

Open paper AskPDF arXiv

Discrete optimization algorithms are often analyzed through continuous-time limiting ODEs, but a convergence certificate for the ODE is not automatically one for the discrete algorithm. We develop contact Hamiltonian systems as a setting where the transfer can be made precise. A contact Hamiltonian

H

J^1(\mathbb{R}^n)

obeys the intrinsic decay identity

\dot H = -H\,\partial_s H

, so an augmented energy

\mathcal{E}

built from

H

, together with the conformal rate

\partial_s H

, is a continuous-time rate certificate whenever

\mathcal{E}

controls the objective gap. Our main theorem states, under three named and independently checkable hypotheses, that an order-

r

contact splitting with step

h

transfers this certificate over the finite horizon set by backward error analysis. The discrete decay envelope is governed by the modified conformal factor up to

O(h^r)

perturbations plus a backward-error shadowing defect, and the mechanism is inherited exactly because the modified Hamiltonian is itself a contact Hamiltonian. Quadratic heavy ball is a fully solvable example: its projected dissipative-leapfrog spectrum agrees with established conformal-symplectic optimization theory, while the augmented contact Hamiltonian yields a sharp objective-to-certificate comparison that verifies the transfer hypotheses. For strongly convex objectives with state-dependent damping, an explicit Bregman-type Lyapunov certificate instead transfers by an auxiliary-shadowing corollary. The decomposition

H=K+V+D

into kinetic, objective-encoding potential, and dissipation terms serves as a design template, with a catalogue of closed-form sub-flows including contact-specific damping families. Numerical experiments confirm the predicted conformal-factor tracking orders and show competitive performance on ill-conditioned benchmarks and deep-learning tasks.

Abstract

HJB Equations

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