MARS-ADMM: Trajectory & Manifold Optimization

• The paper on multi-robot trajectory optimization introduces MARS-ADMM, leveraging stiffness decoupling and log‐barrier penalties to ensure feasibility and rapid convergence under complex constraints.
• The stochastic Riemannian variant applies a single-loop, SARAH-style variance reduction scheme to efficiently solve nonsmooth composite optimization problems on manifolds.
• Empirical results show 10–20× speedups over traditional methods in UAV and robotic arm scenarios, highlighting its potential for decentralized and real-time applications.

MARS-ADMM refers to two distinct but related algorithms based on the alternating direction method of multipliers (ADMM), both designed for challenging settings in optimization. Originally introduced as "Multi-robot ADMM with Robustness and Stiffness Decoupling" for multi-robot trajectory optimization (Ni et al., 2021), it has recently appeared as the first single-loop Stochastic Riemannian ADMM for nonsmooth composite optimization on manifolds (Jin et al., 28 Dec 2025). Both variants leverage structured ADMM splitting, but target different domains—trajectory planning and general composite optimization respectively.

1. Formulation and Problem Settings

The first MARS-ADMM, in multi-robot trajectory optimization (Ni et al., 2021), seeks a solution to a constrained nonconvex optimization problem involving robot trajectories. The primary variables are a sequence of Bézier control points (for UAVs) or configuration variables $\theta\in\R^{\mathrm{DOF}\times N}$ (for robot arms), alongside a uniform time step $\Delta t>0$. The objective functional is a sum of segment-wise costs $\mathcal{O}(X_i,\Delta t)$, potentially including jerk, snap, or time penalties. Collision avoidance is enforced by requiring separating planes between all robot and obstacle convex hulls, with additional kinematic and dynamic constraints: $$\|V_i\|\le v_{\max}\Delta t,\quad \|A_i\|\le a_{\max}\Delta t^2,$$ where $V_i$ and $A_i$ denote velocity and acceleration for trajectory segment $i$. All hard constraints are recast via log-barrier penalties $-\gamma\log(g(\cdot))$ of finite weight, resulting in a fully coupled unconstrained Lagrangian.

The stochastic Riemannian MARS-ADMM (Jin et al., 28 Dec 2025) generalizes the setting to composite nonsmooth problems on a Riemannian manifold $\M\subset\mathbb{R}^n$: $\min_{x\in\M}\;F(x) + g(Ax),$ where $F(x)=\mathbb{E}_{\xi}[f(x,\xi)]$ is a smooth (possibly nonconvex) stochastic objective, $g$ is convex and possibly nonsmooth, and $A$ is a linear operator. The problem can be equivalently rewritten in block-separable form by introducing an auxiliary variable $y$: $\min_{x\in\M, y\in\mathbb{R}^m} F(x) + g(y)\quad \text{s.t.} \quad Ax-y=0.$

2. ADMM Splitting and Stiffness Decoupling

In the multi-robot setting (Ni et al., 2021), MARS-ADMM exploits the distinct numerical properties of cost terms vs. constraint terms. Log-barrier constraints are characterized as stiff due to unbounded gradients near the feasibility region, while the per-segment trajectory cost remains well-conditioned. The algorithm introduces per-segment slacks $\bar{X}_i$ and $\Delta\bar{t}_i$, enforcing $X_i(\theta)=\bar{X}_i$ and $\Delta t=\Delta\bar{t}_i$ via an augmented Lagrangian with penalty parameter $\varrho$: $$\mathcal{L}_{\rm AL} = \sum_i [\mathcal{O}(\bar{X}_i,\Delta\bar{t}_i)] - \gamma\sum_{\text{constraints}}\log(\cdot) + \text{quadratic penalties and dual terms}.$$ ADMM alternation consists of:

• (A) Minimizing stiff terms with respect to $(\theta,\Delta t)$,
• (B) Fast, unconstrained minimization of smooth cost with respect to slacks $(\bar{X}_i,\Delta\bar{t}_i)$,
• (C) Dual ascent for multipliers,
• (D) Alternating minimization on plane parameters.

The stochastic Riemannian MARS-ADMM (Jin et al., 28 Dec 2025) avoids double-loop complexity by integrating all updates into a single loop. It uses momentum-based variance-reduced gradient estimators (SARAH-style recursion) exclusively for the smooth component and leverages constant-size batch sampling.

3. Algorithmic Structure and Key Innovations

The original MARS-ADMM algorithm (Ni et al., 2021) does not update its penalty parameter $\varrho$, mirroring classical ADMM. Stiffness decoupling—keeping log-barrier penalties strictly in the $(\theta, \Delta t)$ subproblem—ensures that slack updates and trajectory cost minimization remain numerically stable and efficient. Separating-plane normals are kept unit-length via exponential map parameterization.

In the stochastic Riemannian context (Jin et al., 28 Dec 2025), each iteration involves:

• Penalty parameter update: $\rho_k = c_\rho k^{1/3}$,
• $y$-update via proximal mapping on $g$,
• Stochastic estimator formation and Riemannian retraction for $x$-update,
• Dual variable and step size updates with adaptive schedules,
• Momentum recursion on gradient estimator, implementing a Riemannian SARAH-style variance reduction.

The momentum-based gradient estimator is: $$v_{k+1} = \nabla f_{\mathcal{S}_{k+1}}(x_{k+1}) + (1-\alpha_{k+1}) \mathcal{T}_{x_{k+1}, x_k}(v_k - \nabla f_{\mathcal{S}_{k+1}}(x_k)),$$ with $\alpha_{k+1} = c_\alpha k^{-2/3}$.

4. Theoretical Guarantees

The multi-robot MARS-ADMM (Ni et al., 2021) exhibits three guarantees:

• Feasibility and Collision-Avoidance: Iterates are maintained strictly feasible by performing line-search shrink-back using continuous-collision-detection, ensuring no constraint violation at convergence.
• Homotopy Preservation: With a strictly feasible initialization, the trajectory cannot cross any obstacle boundaries, so the homotopy class of the path is preserved.
• Convergence: Under mild regularity assumptions (twice-differentiable cost, compact domains, qualified subproblem solvability), the block-coordinate minimization plus dual update scheme converges to a first-order critical point, with primal and dual residuals vanishing in the limit.

The stochastic Riemannian MARS-ADMM (Jin et al., 28 Dec 2025) proves iteration complexity $\tilde{\mathcal{O}}(\epsilon^{-3})$ to reach $\epsilon$-KKT points, matching deterministic ADMM rates and improving on previous stochastic Riemannian operator-splitting bounds ($\mathcal{O}(\epsilon^{-3.5})$). Convergence relies on balancing rates for penalty, primal, dual, and momentum components and employing a merit function that couples expected augmented Lagrangian decrease with control of estimator variance.

5. Decentralization and Implementation Aspects

The first MARS-ADMM (Ni et al., 2021) is implemented in C++11 on Linux, with all per-segment slack and plane updates naturally parallelizable. Parameter choices are uniform, with fixed penalty ($\varrho=0.1$), log-barrier weight ($\gamma=10$), and explicit velocity and acceleration bounds for UAVs and robot arms. Termination is set via $\|\nabla\mathcal{L}\|_\infty$ thresholds, and initialization requires strictly feasible paths (e.g., RRT-connect).

Stochastic Riemannian MARS-ADMM (Jin et al., 28 Dec 2025) avoids double-loop/batch restarts, requiring only a constant number of stochastic gradient evaluations per iteration. Penalty parameters grow slowly with iterations ($\rho_k = c_\rho k^{1/3}$), preventing numerical instability. While theory admits constant-size batches, moderate batch sizes (50-200 samples) offer practical stability.

6. Quantitative Performance and Empirical Comparison

Empirical evaluations for multi-robot MARS-ADMM (Ni et al., 2021) yield:

• Single UAV trajectory planning: Wall-times 34 ms–18.9 s, compared to primal IPM (1,000 ms–420,000 ms), indicating 10–20× speedup; trajectory cost within 0.5–1.5% of primal IPM, often shorter than Safe-Corridor.
• 4-UAV multi-group swaps: 100% success rate vs. 53% for soft-penalty approaches; trajectory length/time 77.8 m/11.3 s vs. 80.1 m/20.6 s.
• 7-DOF KUKA arms swap: RRT path refined from 6.38 m to 1.63 m in 16 s; convergence in ~150 iterations, sub-second per iteration.

Ablation studies show removing stiffness decoupling slows runtime by 3–5× without significant quality change; log-barrier weight $\gamma$ variations ($1$, $10$, $100$) have negligible impact on trajectory quality, but may affect iteration count.

MARS-ADMM in the Riemannian stochastic optimization setting (Jin et al., 28 Dec 2025) achieves near-optimal complexity, closing the theoretical gap between stochastic and deterministic operator-splitting methods.

7. Comparative Analysis and Implications

A summary of performance and theoretical features is presented below.

Variant	Domain	Complexity Bound	Decentralization
Multi-robot MARS-ADMM	Trajectory Opt.	Empirical order-of-mag. speedup	Full parallelization (per segment/plane)
Stochastic Riemannian MARS-ADMM	Composite nonsmooth optimization	$\tilde{\mathcal{O}}(\epsilon^{-3})$	Single loop, constant samples

Both approaches demonstrate that structured nonconvex constraints and composite objectives can be split and solved efficiently via ADMM when carefully coupled with stiffness decoupling and, in the stochastic case, momentum-based variance reduction. This suggests a broad applicability of MARS-ADMM in settings where collision safety, robustness, and decentralized execution are critical, and where optimization problems reside on manifolds or require stochastic or composite objective treatment.

A plausible implication is that the methodological principles from the original multi-robot setting—stiffness decoupling, per-block slack variables, parallelism—are extensible to more abstract or general manifolds, as evidenced by the single-loop stochastic Riemannian MARS-ADMM formulation. The closing of the complexity gap to deterministic methods indicates potential for real-time and large-scale nonsmooth optimization applications under mild regularity assumptions.

References:

"Robust Multi-Robot Trajectory Optimization Using Alternating Direction Method of Multiplier" (Ni et al., 2021) "A Single-loop Stochastic Riemannian ADMM for Nonsmooth Optimization" (Jin et al., 28 Dec 2025)