Feasible Path Algorithms

• Feasible path algorithms are computational methods that generate continuous trajectories while rigorously satisfying geometric, dynamic, and operational constraints.
• They integrate candidate generation, feasibility detection, and local correction modules—using techniques like RRT, convex inner approximations, and local search—to ensure all waypoints remain valid.
• Evaluations focus on metrics such as path optimality, feasibility ratio, and computation time, underscoring their importance in real-time robotic navigation and control applications.

A feasible path algorithm is any computational method designed to construct or ensure the existence of a solution trajectory—typically in the context of motion planning, optimization, or control—that satisfies all imposed constraints (e.g., collision avoidance, dynamic feasibility, operational constraints) at every point along the trajectory. Feasible path algorithms are central in robotics, power system operation, black-box optimization, and program analysis, where ensuring that transitions stay within the admissible set is a primary requirement for safety, reliability, or correctness.

1. Foundational Principles and Definitions

A feasible path, in algorithmic settings, is a continuous or discretized trajectory that links a start (initial) to a goal (target) state while always respecting all problem-specific constraints. These may include:

• Geometric constraints: Collision avoidance with environment obstacles, keeping configurations within the workspace.
• Kinematic/Dynamic constraints: Respecting actuator limits, maximum curvature or turning angles, system dynamics.
• Operational/physical constraints: For example, all points must be AC power flow feasible in electrical networks, or stay within prescribed process boundaries in chemical engineering.

Feasible path algorithms thus go beyond finding simple solutions by enforcing “all-waypoints-feasible” guarantees, often using specialized methods such as local search corrections, hybrid metaheuristic frameworks, convex inner approximations, or predictive constraint satisfaction.

2. Modular Structure of Feasible Path Algorithms

Many feasible path algorithms are organized in modular stages to incrementally generate and correct solution candidates:

Module Type	Role in the Algorithm	Example Technique
Candidate Generation	Produce initial trajectories or waypoints	Metaheuristics (PSO-MFB, RRT)*, Grid/Dijkstra, ML surrogate proposals
Feasibility Detection	Identify infeasible points or segments	Obstacle intersection check, LS module
Correction/Projection	Modify infeasible trajectories to regain feasibility	Local Search, Convex Restriction, Feasibility Iteration
Obstacle/Sensor Integration	Replan in case of dynamic or sensed obstacles	ODA module, Real-time collision avoidance

(*PSO-MFB and RRT refer to Particle Swarm Optimization–Modified Frequency Bat and Rapidly exploring Random Tree algorithms respectively.)

Distinct techniques include:

• Hybrid global-local search: Metaheuristics (e.g., PSO-MFB) generate candidate routes, optionally tuned by local search algorithms that adjust infeasible nodes in response to detected constraint violations (Ajeil et al., 2018).
• Convex feasible set (CFS) approaches: The initial path is systematically corrected, ensuring each segment/waypoint remains within a convex inner approximation of the constraint set, guaranteeing local feasibility (Sun et al., 2019).
• Sequential convex restriction and projection: Solution trajectories iteratively move between convex regions, ensuring each step and all convex combinations remain strictly feasible (Lee et al., 2019).
• Layered/planning architectures: High-level planners propose candidate waypoints/geometric corridors, while low-level controllers or feasibility modules refine the path and ensure constraint satisfaction at the execution level (Csomay-Shanklin et al., 20 Nov 2024).

3. Mathematical Formulation and Optimization Strategies

The central computational goal in feasible path algorithms is to optimize a cost function (e.g., distance, smoothness, energy) over the set of feasible trajectories, expressed mathematically as:

$$\begin{align*} &\min_{\text{path}}~ J(\mathbf{p}) \ \text{s.t.}~~ & C_i(\mathbf{p}_k) \leq 0,~~\forall k,~\forall i \end{align*}$$

where $\mathbf{p}_k$ denotes the $k$-th waypoint or state of the discretized path, and $C_i$ are the constraint functions (e.g., obstacle avoidance, kinematic bounds).

Standard methods are:

• Metaheuristic-based optimization: Swarm-based or evolutionary algorithms optimize a multi-objective fitness (weighted sum of objectives such as shortest distance and path smoothness) and are augmented with correction steps to recover feasibility if waypoints enter forbidden regions (Ajeil et al., 2018).
• Convexification and inner approximation: The feasible set is conservatively approximated (via polytopes, convex restrictions), and optimization occurs within this inner region, ensuring feasibility by construction (Lee et al., 2019, Turizo et al., 5 Aug 2024).
• Feasibility-preserving sequential updates: In sequential linear/quadratic programming frameworks, after each linearized subproblem, a correction or projection phase ensures the update is feasible for the original nonlinear constraints before accepting the step (Kiessling et al., 2022, Ma et al., 16 Feb 2024).
• Adaptive sampling and surrogate modeling: For black-box functions or simulation-based settings, adaptive pre-filtering and the construction of differentiable ML surrogates allow feasible-path-seeking algorithms to be applied even when the direct form of the constraints is unknown (Zhang et al., 25 Sep 2025, Zhang et al., 29 Jan 2025).

4. Treatment of Infeasibility and Correction Mechanisms

A critical aspect is the detection and handling of infeasible intermediate points. Corrective strategies include:

• Local Search Correction: If a candidate waypoint is infeasible (e.g., inside an obstacle or between nodes traverses an obstacle), local search computes a corrected location just outside the forbidden region, using the relative geometry of obstacles and a safety margin (Ajeil et al., 2018).
• Projection/Feasibility Iteration: After each major update in an optimizer (e.g., in FSLP), a secondary subproblem projects the candidate back onto the nonlinear feasible set using only evaluations of the constraint functions, not their derivatives (Kiessling et al., 2022).
• Barrier/Slack Variables: In process optimization, infeasible QP or LSQ subproblems are regularized with slack variables (supporting relaxation) and hybrid switching between formulations to recover feasible search directions in the presence of noise or nonconvexity (Ma et al., 16 Feb 2024).

5. Quantitative Metrics and Empirical Performance

Performance of feasible path algorithms is typically assessed on:

• Path optimality: Comparison of total path length (e.g., in meters), path smoothness (aggregated angle or curvature deviations), or in cost values (dispatch cost, energy, risk).
• Feasibility Ratio: For probabilistic/convex relaxation methods, the percentage of paths generated that are strictly feasible under all constraints (Sun et al., 2019).
• Computation time: Time to convergence, with sub-second performance on mid-size robotic environments and under 2 seconds for optimization-based system design tasks (Zhang et al., 29 Jan 2025).
• Scalability: Capability to maintain real-time operation or sublinear query times with respect to problem size, e.g., $O(\log n)$ query time for feasibility in rectilinear environments (Fan et al., 15 Apr 2025).
• Robustness and reliability: Empirical studies (e.g., over benchmarks) showing 100% solution rate in the presence of dense obstacles or stiff constraints (as in grid-based plus path-smoothing algorithms (Sun et al., 2019)).

Empirical studies frequently compare feasible path algorithms against alternatives such as A*, artificial bee colony (ABC) methods, or conventional metaheuristics, with feasible path methods demonstrating either improved success rates, smoother/shorter paths, or superior time-to-solution (Ajeil et al., 2018, Sun et al., 2019, Huang et al., 2022).

6. Applications and Impact Across Domains

Feasible path algorithms are broadly applicable. Major application domains include:

• Autonomous robotic navigation: Real-time motion and trajectory planning in dynamic or cluttered environments, including UAV collision avoidance and warehouse AGV routing (Ajeil et al., 2018, Divkoti et al., 2019).
• Power system operation: Constructing explicit feasible transition paths between operating points in nonconvex AC optimal power flow regions, providing safety guarantees for real-time control actions (Lee et al., 2019, Turizo et al., 5 Aug 2024).
• Black-box and simulation-based optimization: Enabling KKT-convergent algorithms in settings where objective and constraint functions are only accessible via expensive simulation or ML surrogates (Zhang et al., 25 Sep 2025, Zhang et al., 29 Jan 2025).
• Cyber-physical layout design: Minimizing transportation costs in facility or chip layout using door-to-door feasible path measurements that respect placement and corridor constraints (Shavarani et al., 17 Apr 2025).
• Program analysis: Data flow analyses leveraging feasible path maximal fixed-point solutions to filter infeasible execution paths in control flow graphs, improving static program correctness and reducing false alarms (Pathade et al., 2022).
• Multi-agent system routing: Polynomial-time feasibility checking and solution construction in multi-agent pathfinding on strongly connected digraphs, including warehouse or industrial robot fleets (Ardizzoni et al., 2022).
• Real-time and adaptive control: Layered frameworks that combine high-level feasible path planning with low-level robust tracking to guarantee dynamic feasibility for legged and hopping robots in unpredictable, cluttered environments (Csomay-Shanklin et al., 20 Nov 2024).

7. Comparative Strengths and Limitations

• Strengths: Feasible path algorithms deliver strong guarantees of constraint satisfaction at all intermediate states and often show improved robustness in uncertain, high-consequence, or real-time applications. Modular architectures allow for the integration of advanced optimization, machine learning, and geometric correction steps.
• Limitations: Increased computational complexity is possible, especially when feasibility correction or projection steps are expensive, or the feasible region is highly nonconvex or disconnected. Conservative (inner) approximations can lead to suboptimal (though safe) solutions. For complex, high-dimensional surrogates, efficient derivative computation remains a practical challenge, though advances in automatic differentiation and hybrid relaxations offer solutions (Ma et al., 16 Feb 2024, Zhang et al., 29 Jan 2025).

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In sum, feasible path algorithms represent an overview of optimization, geometric reasoning, constraint enforcement, and—where relevant—learning-based surrogate modeling, providing a principled foundation for ensuring safe and effective path generation in a variety of challenging real-world and computational environments. Their ongoing development remains an active area of research at the interface of robotics, optimization, control, and applied computer science.