Convexified Trajectory Matching

• Convexified Trajectory Matching is a set of optimization and learning techniques that transform nonconvex trajectory problems into convex formulations using domain-specific relaxations.
• It employs methods such as buffered Voronoi cells, convex polytopic decompositions, and convex surrogates to guarantee safety, convergence, and computational efficiency.
• Applications span robotics, autonomous systems, and dataset distillation, offering robust performance improvements and tighter optimality guarantees in dynamic environments.

Convexified Trajectory Matching (MCT) refers to a class of optimization and learning methodologies that recast inherently non-convex trajectory or path-matching problems as convex, or sequence-of-convex, programs. By leveraging domain-specific relaxations, problem reformulations, and convex surrogates, MCT enables globally optimal or provably convergent solutions with significant improvements in computational efficiency, solution reliability, and safety guarantees across a range of fields including robotics, autonomous systems, optimal control, and modern dataset distillation.

1. Theoretical Motivation and Problem Statement

The foundational challenge addressed by Convexified Trajectory Matching is the non-convexity arising in trajectory planning, matching, or tracking in high-dimensional dynamic systems—whether due to nonlinear agent dynamics, collision-avoidance, time-varying targets, or matching of learning dynamics in synthetic data. MCT frameworks seek to identify problem representations or relaxations under which the feasible trajectory space can be tightly approximated or covered by convex sets, or for which convex surrogate losses can enforce the required trajectory matching properties.

The essential principle of MCT is the projection or relaxation of the nonlinear, non-convex constraints of the original problem onto “convexified” sets or paths, which may be realized as:

• Graphs of convex sets in space-time (Philip et al., 2024)
• Convex polytopic or polyhedral cells for agent safety (Dickson et al., 2024)
• Buffered Voronoi Cells (BVC) for distributed collision avoidance (Chen, 2024)
• Convex surrogates in policy or parameter space, including straight-line “convexified” paths in neural network weights (Zhong et al., 2024)
• Linear or convex quadratic surrogates for multi-affine or bi-affine constraints (Singh et al., 2019)
• Higher-order manifolds induced by Taylor expansions under monomial coordinates (Regantini et al., 12 May 2025)

The general idea is that, under such convexifications, optimal or feasible trajectory matching can be achieved through powerful convex optimization algorithms—such as SOCP, SDP, QP, or consensus-ADMM.

2. Representative Formulations and Relaxations

Diverse domains instantiate MCT by problem-specific convexification schemes:

Vehicle Routing with Moving Targets

In “A Mixed-Integer Conic Program for the Moving-Target Traveling Salesman Problem based on a Graph of Convex Sets” (Philip et al., 2024), time-windowed straight-line target trajectories are recast as convex line segments $X_i \subset \mathbb{R}^3$ in space-time. The full path-finding problem reduces to a shortest path in a graph whose node-sets are convex, with relaxed perspective-cone constraints yielding a convex conic program whose relaxation tightly bounds the integer optimum. This yields substantially improved optimality gaps and runtime.

Multi-Robot Distributed Coordination

In “Multi-Robot Trajectory Generation via Consensus ADMM: Convex vs. Non-Convex” (Chen, 2024), classical minimum-distance collision constraints are replaced by affine constraints corresponding to Buffered Voronoi Cells that are convex polytopic regions moving over time. Embedded within a consensus-ADMM framework, all iterates are feasible for collision and dynamics at every step, guaranteeing safety and convergence.

Spline Trajectory Tracking and Obstacle Avoidance

In “Spline Trajectory Tracking and Obstacle Avoidance for Mobile Agents via Convex Optimization” (Dickson et al., 2024), the workspace is partitioned into overlapping convex cells. Polynomial trajectories are tracked via output-feedback controllers whose convergence and safety (Control Lyapunov and Control Barrier Functions) are enforced by a series of LMIs, solved as SDPs for each cell. The per-cell controller switching schedule enables globally robust and safe trajectory matching.

Dataset Distillation via Convexified Trajectory

In “Towards Stable and Storage-efficient Dataset Distillation: Matching Convexified Trajectory” (Zhong et al., 2024), the matching objective is convexified by projecting the noisy expert parameter trajectory onto a straight-line convex combination between initial and terminal weights. This yields not only more stable and memory-efficient student matching but also accelerated convergence rates.

Rendezvous and Spacecraft Guidance

In “Convex Trajectory Optimization via Monomial Coordinates Transcription for Cislunar Rendezvous” (Regantini et al., 12 May 2025), nonlinear dynamics are approximated by high-order monomial-coordinate expansions via Differential Algebra. The original non-convex manifold constraints are linearized, and Sequential Convex Programming alternates projections and convex SOCP solves, yielding robust guidance solutions with minimal online computation.

3. Algorithmic and Computational Methodologies

Most MCT approaches share a generic algorithmic structure:

• Convexification of Constraints/Trajectories: Non-convex sets (e.g., trajectories, safety regions, parameter paths) are replaced or tightly covered by convex sets or combinations, preserving the essential geometric or dynamical properties relevant for matching.
• Discrete or Distributed Decomposition: Many formulations exploit time or space decomposition, enabling per-segment convex programming (e.g., overlapping spatial cells, timestep-wise QP blocks, local copies in consensus-ADMM).
• Iterative Convex Program Solving: The primary optimization loop involves solving convex subproblems, either globally (e.g., MI(SOCP)/SDP, QP) or distributedly (C-ADMM), potentially with consensus or trust-region updates when linearizations are required.
• Projection and Surrogate Updates: For cases involving nonlinear manifolds or policies (e.g., monomial coordinates, angular variables), convex surrogates (e.g., 1D angle projections) or linearized constraints are employed as substitutes for non-convex steps (Singh et al., 2019, Regantini et al., 12 May 2025).
• Switching/Concatenation: For environments with overlapping convex coverage or mode switching, the solution is pieced together via controller/schedule switching, ensuring global feasibility (Dickson et al., 2024).

The table below summarizes representative MCT formulations across selected domains:

Domain	Convexification Strategy	Solving Method
Moving-target VRP (Philip et al., 2024)	Convex sets in graph nodes	MI(SOCP)/conic program
Multi-robot (Chen, 2024)	Buffered Voronoi Cells (BVCs)	Consensus-ADMM, QP
Obstacle avoidance (Dickson et al., 2024)	Convex workspace decomposition	LMI/SDP per cell
Dataset distillation (Zhong et al., 2024)	Straight-line convex path in param	Gradient-based opt
Rendezvous (Regantini et al., 12 May 2025)	Monomial manifold linearization	SCP, SOCP

4. Theoretical Guarantees and Performance Analyses

MCT methods, by construction, often inherit strong guarantees from convex program theory:

• Feasibility and Safety: By exactly enforcing convex constraints at each step (e.g., BVCs, CBF-defined safe cells), trajectories (including intermediate ADMM iterates or offline trajectories) remain feasible and safe (Chen, 2024, Dickson et al., 2024).
• Convergence Rates: For linearly constrained QPs (e.g., distributed consensus-ADMM), global and linear convergence rates are provable (Chen, 2024). In dataset distillation tasks, the uniform direction of descent in convexified parameter space yields faster or linear convergence (Zhong et al., 2024).
• Tightness of Convex Relaxation: Compared to standard “big-M” relaxations, perspective-cone relaxations and GCS constructions provide much tighter lower bounds and narrower optimality gaps (e.g., 40–60% tighter for MT-TSP (Philip et al., 2024)).
• Robustness: Bounded disturbance rejection is observed empirically when offline margins or slack variables are used as part of the convex program (Dickson et al., 2024).
• Computational Performance: MCT formulations routinely admit greater computational efficiency, e.g., orders-of-magnitude faster solution times in MI(SOCP) (Philip et al., 2024), 2–3× faster ADMM convergence (Chen, 2024), reduction to 8% storage costs in dataset distillation (Zhong et al., 2024), or 2–5× CPU time reduction in onboard cislunar rendezvous (Regantini et al., 12 May 2025).

5. Applications and Empirical Results

The MCT paradigm has been instantiated and validated in a range of domains.

• Vehicle Routing: MT-TSP instances up to n=20 targets saw order-of-magnitude speedup and optimality gap reductions via GCS-based MI(SOCP) (Philip et al., 2024).
• Multi-Agent Planning: Convex BVC-ADMM enables fully distributed, collision-free planning, with strict reduction in infeasibility and objective cost, as demonstrated for 3–5 robot systems requiring hundreds to thousands fewer ADMM iterations compared to non-convex baselines (Chen, 2024).
• Safety-Critical Tracking: Spline/CELL-MCT controllers synthesized by SDP guarantee smooth transitions and collision-safe tracking across arbitrary nonconvex polygonal environments (Dickson et al., 2024).
• Dataset Distillation: Convexified matching of parameter trajectories increases test accuracy on CIFAR-10/100 and Tiny-ImageNet by 0.8–2.4 percentage points, with 20–50% fewer training steps required and a reduction to 8% of the storage requirements of strict MTT (Zhong et al., 2024).
• Space Guidance: Monomial SCP in cislunar rendezvous yields high-accuracy, low-computation (few seconds per trajectory on commodity CPUs), and outperforms classical linear STM approaches, with guidance errors below ~10⁻³ km (Regantini et al., 12 May 2025).

6. Extensions and Generalization

The convexification techniques underlying MCT extend to a variety of problem structures:

• Higher-order or piecewise-polynomial target or reference sets for trajectory generation (Philip et al., 2024, Dickson et al., 2024)
• Stochastic or robust versions via belief-state tubes or stochastic DA expansions (Philip et al., 2024, Regantini et al., 12 May 2025)
• Time-dependent modalities and hybrid transport (multi-modal, varying-speed frameworks) (Philip et al., 2024)
• Flexible convex cell decompositions and sparse state representations for efficient hardware or memory-constrained implementations (Dickson et al., 2024, Regantini et al., 12 May 2025)
• Broader learning frameworks, where learning objectives can be convexified by trajectory relaxation, as in modern dataset distillation (Zhong et al., 2024)

A plausible implication is that continued advances in convexification strategies and distributed convex optimization algorithms will further expand the applicability of MCT in domains where safety, reliability, and computational load are paramount.

7. Limitations and Open Challenges

• Tightness vs. Expressivity: While convex relaxations provide provable lower bounds and feasible safety guarantees, there may remain solution gaps for highly non-convex or discontinuous tasks. The success of MCT often depends on the quality of the convex surrogate.
• Scalability and Parallelization: Distributed MCT variants (e.g., consensus-ADMM) scale well in sparse settings, but communication and asynchrony in large agent networks may introduce complexity.
• Nonlinear Dynamics and Model Uncertainty: Higher-order manifold convexification (e.g., in monomial SCP) requires careful management of truncation error regions and may demand reparameterization for broad initial conditions (Regantini et al., 12 May 2025).
• Control Synthesis under Switching and Disturbance: Ensuring seamless controller switching and robust performance across cell boundaries remains nontrivial in high-dimensional, disturbance-prone environments.

The systematic convexification and matching of trajectories in control, optimization, and learning settings thus continues to be an active area for theoretical research and practical innovation.