Trajectory-Based Optimization Algorithm

• Trajectory-based optimization algorithms are methods that use continuous or discretized trajectories driven by dynamical systems to search for local or global minima.
• They combine projected and quotient gradient approaches to ensure feasibility and effectively explore complex, nonconvex landscapes.
• Implementations in multi-agent systems, robotics, and reinforcement learning show improved convergence and scalability through parallelization and operator-splitting techniques.

Trajectory-based optimization algorithms constitute a versatile class of methods for solving constrained and unconstrained optimization problems where the solution is characterized as a continuous or discretized trajectory. These algorithms are central in fields such as optimal control, robotics, machine learning, and signal processing. The unifying principle is that optimization is performed by following continuous trajectories—often defined by differential equations or discrete iterations—within the feasible space, searching for local or global minima under possibly complex, high-dimensional constraints.

1. Mathematical Foundations and Core Principles

The mathematical backbone of trajectory-based optimization lies in formulating the search for optima as the evolution of points in parameter space according to dynamical (typically gradient-driven) systems. For constrained problems, this formulation leads naturally to use of projected and quotient dynamics:

• Projected Gradient System (PGS): Optimization constrained to a smooth manifold $M = \{x : h(x) = 0\}$ is performed by the ODE:

$\dot x = -P(x) \nabla f(x), \quad x \in M$

where $P(x)$ projects onto the tangent space of $M$. This ensures trajectories remain feasible at all times. Local minima (i.e., constraint-satisfying KKT points) are asymptotically stable equilibria under this flow (Khodabandehlou et al., 2018).

• Quotient Gradient System (QGS): Exploration of disconnected feasible space components is achieved via:

$\dot x = -D h(x)^T h(x)$

which acts as a steepest descent for the infeasibility measure $\|h(x)\|^2$. Each connected component of the feasible set is a locally attracting equilibrium manifold (Khodabandehlou et al., 2018).

The dual application of these systems enables the systematic identification of all local minima for problems where the feasible region may be disconnected or highly nonconvex.

2. Algorithmic Structures and Global Search Strategies

Trajectory-based optimization methods exhibit significant structural variation depending on problem class. Key variants include:

• Dynamical Two-Phase (DTP) Methods: As described for RNN training (Khodabandehlou et al., 2018), these alternate between the QGS (feasibility-seeking) and PGS (local minimization) phases to locate multiple minima across feasible components, iteratively escaping basins of attraction to fully explore the solution landscape.
• Operator-splitting and ADMM-based Trajectory Exploration: Sequential operator-splitting with consensus ADMM, as seen in OS-SCP (Ganiban et al., 18 Nov 2025), frames exploration as a parallelized agent system where diverse initializations are coordinated via consensus dynamics. Each agent solves a local convexified subproblem (via sequential convex programming), with consensus enforced via augmented Lagrangian terms. This enhances exploration and the capacity to find better local minima than single-trajectory SCP.
• Parallel and Decentralized Trajectory Optimization: For large-scale or multi-agent systems, trajectory optimization can be decomposed into parallelizable subproblems, solved either via consensus-ADMM with per-segment closed-form quadratic programs (e.g., TOP (Yu et al., 14 Jul 2025)) or using decentralized QP-based receding-horizon schemes in multi-robot settings (R et al., 2018, Krishnan et al., 2019).
• Policy Search and Reinforcement Learning Approaches: In model-free or partially unknown dynamical settings, trajectory-based policy optimization is performed via trust-region methods with explicit KL-divergence constraints (e.g., MOTO (Akrour et al., 2016)), or via curriculum-guided RL leveraging reference trajectories (Ota et al., 2019).

3. Stability, Convergence, and Theoretical Guarantees

Rigorous Lyapunov-based proofs confirm stability and convergence properties for the main classes of trajectory-based algorithms:

• PGS/QGS-based Methods: Lyapunov functions $V_Q(x) = \frac12 \|h(x)\|^2$, $V_P(x) = f(x)$ decrease strictly along trajectories except at critical points, ensuring that feasible sets are attracting and constraint-satisfying minima are stable equilibria (Khodabandehlou et al., 2018).
• Monotonic Improvement and Trust-Region Updates: Model-free approaches with exact KL constraints (e.g., MOTO) guarantee monotonic expected policy improvement under local quadratic-Q-function regression, with explicit bounds derived from performance-difference identities and Pinsker-type arguments (Akrour et al., 2016).
• Recursive Feasibility and Convergence in Receding Horizon Schemes: Two-step optimization-based receding horizon planners (Bergman et al., 2019) guarantee that cost-to-go decreases monotonically and feasibility is preserved at each iteration, leading to finite-time convergence under running-cost positivity conditions.
• ADMM and SCP Operator-Splitting: In convex settings, ADMM-based consensus splitting provides provable convergence; for nonconvex trajectory spaces, convergence to stationary points can be demonstrated under regularity conditions (Ganiban et al., 18 Nov 2025).

4. Implementation Methods and Computational Considerations

Implementation details in recent literature reflect key trade-offs between computational complexity, parallelizability, trajectory smoothness, and constraint adherence:

• Projection and Linearization Overheads: Each PGS and QGS integration step involves Jacobian and projection computations, with per-step cost dominated by small-to-moderate linear solves (Khodabandehlou et al., 2018). Per-segment QP solves in consensus-ADMM yield constant per-iteration time when perfectly parallelized (Yu et al., 14 Jul 2025).
• Parallelization: Frameworks such as TOP exploit massive parallelization (N-core CPU or GPU) to achieve per-iteration O(1) scaling with respect to trajectory segmentation (Yu et al., 14 Jul 2025).
• Handling Constraints: Hard constraints are embedded via slack variables and equality constraints (PGS/QGS), barrier or penalty terms (SCP-based or interior-point lower levels (Howell et al., 2021)), or sequential constraint evaluation (robust MOEA with PCE (Takubo et al., 2022)).
• Real-Time and Decentralized Execution: Receding-horizon updates, receding B-spline-based QPs, and decentralized rollout architectures (R et al., 2018, Krishnan et al., 2019) allow real-time multi-agent trajectory planning.

5. Applications and Performance Benchmarks

Trajectory-based optimization algorithms enable high-performance solutions to diverse control, inference, and planning tasks:

• Recurrent Neural Network Training: The DTP method surpasses both genetic algorithms and classic error-backpropagation in sum-of-squares error and generalization across NARMA, nonlinear, and hysteresis identification tasks (Khodabandehlou et al., 2018).
• Large-Scale Trajectory Smoothing: The TOP framework achieves greater than tenfold speedup over state-of-the-art GCOPTER on dense trajectory smoothing, maintaining or improving trajectory cost and smoothness in real-world quadrotor trials (Yu et al., 14 Jul 2025).
• Multi-Robot and Multi-Agent Systems: Decentralized receding horizon planners based on polynomial or B-spline representations solve collision-free multi-robot navigation with real-time rates, low computational latency, and high (>99%) success rates across up to 13 robots (Krishnan et al., 2019, R et al., 2018).
• RL-based Trajectory Planning: RL agents trained with trajectory-based reward shaping and curriculum learning exhibit superior path efficiency, smoothness, and goal generalization compared to PID and pure RL baselines (Ota et al., 2019).
• Swarm UAVs and Online Optimization: Distributed PE-PSO (persistent exploration PSO) with entropy adaptation and task-allocation achieves real-time collision-free multi-target assignment with significant reductions in planning time versus previous swarm algorithms (Li et al., 18 Jul 2025).
• Robust Multi-Objective Control: PCE-constrained MOEA delivers robust open-loop control policies for stochastic flight control, efficiently handling chance constraints and yielding Pareto sets robust to environmental uncertainty (Takubo et al., 2022).

6. Extensions, Recent Directions, and Limitations

Current research in trajectory-based optimization focuses on enhancing scalability, robustness, and global search capabilities:

• Global Optimization and Quantum Algorithms: Exact discretization-based trajectory search allows for application of quantum global-search (Grover-based) algorithms, achieving quadratic reductions in oracle complexity relative to classical search at a given solution fidelity (Shukla et al., 2019).
• Bayesian and Multi-Objective Optimization: GP-based trajectory modeling combined with multi-objective Bayesian acquisition (such as TEHVI) enables efficient identification of epoch-aware trade-offs in hyperparameter optimization with built-in early stopping (Wang et al., 24 May 2024).
• Koopman-Based Partial Convexification: Trajectory optimization with mixed boundary constraints uses a Koopman lift to convexify the high-dimensional lower-level dynamics while restricting nonconvexity to a small upper-level problem over boundary parameters and terminal time (Abou-Taleb et al., 4 Dec 2024).
• Exploration of Nonconvex Landscapes: Operator-splitting SCP frameworks enhance exploration and overcome local-trap sensitivity via agent-based ADMM consensus dynamics, yielding higher-reliability global minima for nonconvex path-planning tasks (Ganiban et al., 18 Nov 2025).

Notable limitations include potential computational overhead for large-scale outer global searches (Das et al., 2023), reliance on convexity in the inner problems, and the need for high-quality feasibility projections or constraint relaxations in highly nonconvex spaces. However, with careful construction and parallel hardware exploitation, trajectory-based algorithms remain state-of-the-art tools for continuous optimization under complex structural and dynamical constraints.