Locally Optimal Obstacle Displacements

• The paper presents a framework where locally optimal obstacle displacements satisfy KKT conditions, ensuring first-order stationarity and second-order sufficiency.
• It details algorithmic strategies such as two-stage overlap+displacement and horizon slicing that balance trajectory quality with minimal physical or functional adjustments.
• Empirical results demonstrate practical trade-offs: higher displacement penalties yield longer paths with fewer overlaps, while efficient convergence is achieved in both robotic and PDE-constrained settings.

Locally optimal obstacle displacements are a class of solutions in motion planning, control, and variational problems involving the adjustment of obstacles or constraints to facilitate feasible robot behavior or to improve system performance. The concept encompasses both path planning with movable obstacles (where physical objects are displaced) and obstacle control for PDE or variational-inequality systems (where obstacles represent functional constraints). Locally optimal solutions are characterized by their satisfaction of first-order stationarity and second-order sufficiency conditions, or by representing Karush–Kuhn–Tucker (KKT) points of an appropriate constrained optimization formulation. This article surveys the mathematical formulations, algorithmic strategies, and theoretical properties of locally optimal obstacle displacement in several domains.

1. Mathematical Formulations Across Domains

Obstacle displacement arises in multiple contexts. In robot motion planning, the minimum obstacle displacement (MOD) and minimum constraint displacement (MCD) problems are formulated as mixed-integer (or continuous) nonlinear programs where robot and obstacle states are decision variables. In PDE-constrained optimization, obstacle displacements correspond to modifications of lower and upper obstacle functions in variational inequalities (VIs).

For motion planning, canonical formulations include: $$\min_{x_{1:T},\,\delta} \quad w^x \sum_{k=1}^{T-1} \|x_{k+1}-x_k\| + w^d\sum_{i=1}^n \|\delta^i\|$$ subject to

$$x_{k+1} = f(x_k, u_k), \qquad x_k \notin \bigcup_{i=1}^n (o^i+\delta^i)$$

where $x_{k}$ is the robot state, $o^i$ is the original obstacle position, and $\delta^i$ is its displacement (Thomas et al., 2023).

In the variational-inequality context,

$\min_{(\phi,\psi)\in U_{ad}} J(\phi,\psi)$

where $y = y(\phi,\psi)$ solves the bilateral obstacle VI: $$a(y, v-y) \ge \langle f, v-y\rangle \quad \forall v \in K(\phi,\psi) = \{v: \phi \le v \le \psi\}$$ with $J(\phi,\psi)$ typically a tracking-type cost (Ghanem et al., 2015).

Constraint displacement problems further generalize to allowing controls on both path and obstacles, minimizing joint costs on path quality and displacement (Thomas et al., 15 Nov 2025).

2. Optimization Landscape and Local Optimality Conditions

Locally optimal obstacle displacements are formally characterized by KKT conditions of the respective nonconvex programs. In PDE-constrained settings, the non-smooth nature of the obstacle-to-state map is handled via penalization (e.g., Moreau–Yosida smoothing) leading to semilinear state equations whose derivatives enable first and second-order conditions (Ghanem et al., 2015). For motion planning formulations, the nonconvexity stems from collision avoidance; the locally optimal solution is a joint local minimizer with respect to both path and displacement variables.

Explicitly, for the penalized obstacle VI,

• First-order stationarity: gradients of penalized cost with respect to obstacle variables vanish.
• Second-order sufficiency: Hessian along admissible directions remains positive definite (Ghanem et al., 2015).

For motion planning with overlap-based displacement (e.g., two-stage overlap + displacement (Thomas et al., 15 Nov 2025)), local optimality is defined by the property that neither path nor displacement variables can be infinitesimally perturbed to reduce cost while preserving feasibility to first order.

3. Computational Methods and Algorithmic Strategies

Several algorithmic frameworks address locally optimal obstacle displacement:

A. Two-Stage Overlap + Displacement

• Stage 1 "Overlap": Compute a path trading off trajectory quality and overlap-penalized cost, via nonlinear model-predictive control (MPC) or sequential programming.
• Stage 2 "Displacement": With the path fixed, solve a reduced-size nonlinear program per obstacle to find minimum-magnitude displacements that remove all path-obstacle overlaps, subject to non-intersection constraints. Analytic solutions are available for disk robots and obstacles; more generally, local optimization (e.g., interior-point) is used (Thomas et al., 15 Nov 2025).

B. Mixed-Integer Programming and Approximation via Horizon Slicing

• The MOD problem is naturally formulated as a MIQP and solved exactly for small instances.
• For large problems, the "horizon slicing" heuristic solves smaller MIQPs over shorter time intervals, producing concatenated locally optimal solutions with explicit approximation guarantees; computational complexity is vastly reduced at the cost of a bounded suboptimality (Thomas et al., 2023).

C. PDE/VIs with Smooth Penalty

• The bilateral obstacle problem is discretized (e.g., finite difference), and the penalized first-order optimality system is solved using Gauss–Seidel–Newton iterations. Explicit formulas for gradients and Hessian-vector products yield efficient, convergent updates (Ghanem et al., 2015).

4. Theoretical Guarantees and Complexity

The underlying optimization problems are nonconvex and, in general, NP-hard (Thomas et al., 2023). The two-stage approaches and horizon slicing provide the following theoretical features:

• Local optimality: Solutions are KKT points for their respective formulations, under standard smoothness and feasibility assumptions (Thomas et al., 15 Nov 2025).
• Approximation guarantees: For horizon-slicing, the concatenated solution's cost $J^s$ satisfies $J^s \leq mJ^*$, where $m$ is the number of slices and $J^*$ is the global optimum (Thomas et al., 2023).
• Feasibility: Displacement-stage local programs are always (in principle) feasible if unbounded obstacle motion is allowed, but practical robustness requires good initialization and geometric regularity (Thomas et al., 15 Nov 2025).
• Convergence: Gauss–Seidel–Newton schemes for PDE formulations exhibit linear-to-superlinear convergence after entering the active-set regime (Ghanem et al., 2015).

5. Practical Implementation and Numerical Performance

Locally optimal obstacle displacement methods have been validated across several platforms and settings:

• In the two-stage overlap+displacement framework, benchmarks in environments with 53 circular obstacles or 19-polygonal settings yield total displacement costs that strictly decrease as overlap weights increase. Second-stage solution times are approximately 1 second per obstacle; full runs complete in seconds using standard solvers (Thomas et al., 15 Nov 2025).
• In the MOD horizon-slicing approach, moderate slice counts ($m=2,3$) yield near-optimal solutions at dramatically lower computational cost. Empirical cost gaps align with theoretical $m$-approximation, and path/displacement trade-offs are straightforwardly tuned via weighting parameters (Thomas et al., 2023).
• Finite-difference PDE discretizations with 200 grid points, relaxation parameter $0.75$, and small regularization drop the residual below $10^{-8}$ within 50 iterations, demonstrating convergence to locally optimal obstacles and state profiles (Ghanem et al., 2015).

Empirical path and displacement costs exhibit the expected qualitative trade-offs—higher displacement penalties induce longer but less intrusive paths, and vice versa. Active-set dynamics in the optimization algorithms yield rapid residual and cost reduction once the correct contact sets are identified.

6. Applications and Extensions

Recent research subsumes classical problems (minimum constraint displacement, removal, and navigation among movable obstacles—NAMO), manipulation-under-clutter, and PDE-constrained design. The formalism is general: by appropriate choice of the overlap cost transform $h$ (e.g., linear, saturating), the framework covers a variety of geometric, combinatorial, and physical objectives (Thomas et al., 15 Nov 2025).

Extensions include:

• Real-time MPC implementations for dynamic settings.
• Integration with range sensors to address partially known or dynamic environments (Cheniouni et al., 29 Dec 2024).
• Robust heuristics for degenerate geometric cases and initialization failures.
• Anytime/online variants combining horizon-slicing and incremental planning (Thomas et al., 2023).

A plausible implication is that the emphasis on local optimality—via explicit stationarity conditions, approximate dynamic programming, or constrained continuous optimization—enables computationally efficient solutions in classes of constraint-modification problems that are otherwise intractable globally. This suggests a unifying methodology for both physical-object displacement (motion planning) and functional-constraint manipulation (PDE, VI control).