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REVIEW 2 major objections 4 minor 49 references

A biconvex method for minimum-time motion planning through sequences of convex sets

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A biconvex alternating scheme solves minimum-time motion planning through convex sets, returning complete, anytime, near-optimal trajectories without tuning.

desk verdict SCS is a genuinely useful biconvex planner with transparent math and honest limitations; the strict-positivity assumption is real but disclosed, and the paper deserves a serious referee. read the letter →

arxiv 2504.18978 v1 pith:UQ5B65WR submitted 2025-04-26 cs.RO

Tobia Marcucci , Mathew Halm , Will Yang , Dongchan Lee , Andrew D. Marchese This is my paper · ORCID
classification cs.RO
keywords motionplanningminimum-timetrajectorybiconvexoptimizationconvexrestrictionBéziercurvessequenceofsetsanytimealgorithm
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to show that minimum-time motion planning through a fixed sequence of convex sets, a nonconvex problem because trajectory shape and timing are coupled, can be solved reliably by alternating two convex subproblems. The proposed method, SCS, keeps either the transition points or transition velocities fixed at each step, and each subproblem is a conservative convex approximation of the original problem, so every iteration produces a feasible trajectory whose duration is no worse than before. The authors claim this makes the algorithm complete (it always finds a feasible trajectory), anytime (a feasible trajectory is available if stopped early), and free of line-search or trust-region parameters. On benchmarks, SCS returns trajectories within roughly 1–3% of the duration found by state-of-the-art nonconvex solvers in a fraction of their runtime, and it designs lower-duration plans than waypoint and trust-region alternatives on a warehouse package-transfer task. If these claims hold, robot systems that need smooth, fast, reliable motions can avoid brittle solver tuning and feasibility failures in online operation.

What carries the argument

The carrying mechanism is a pair of convex restrictions of one nonconvex program. In the fixed-transition-velocity subproblem, velocity continuity becomes linear and the acceleration set $T_i^2\mathcal{A}$ is replaced by the tangent underestimate $\bar T_i(2T_i-\bar T_i)\mathcal{A}$ around the nominal traversal time. In the fixed-transition-point subproblem, the trajectory inside each set is reparameterized by reciprocals $S_i=1/T_i$ and scaled position functions $r_i=q_i/T_i$, which makes velocity constraints convex and leaves the acceleration constraint as the only nonconvexity; it is then relaxed by the analogous underestimate $\bar T_i(2-\bar T_i S_i)\mathcal{A}$. Each subproblem is feasible whenever the current trajectory is feasible and yields a trajectory of no greater duration, which is what makes full-step alternation, monotone convergence, completeness, and the anytime property possible. Bézier curves with their convex-hull property supply the finite-dimensional implementation.

What would settle it

Take a one-dimensional or two-dimensional corridor satisfying Assumption 1 whose true time-optimal trajectory is known from dynamic programming or bang-bang analysis; run SCS with high Bézier degree and a tight tolerance (for example $\varepsilon=10^{-4}$) and measure the duration gap and whether every early-stopped iterate satisfies all constraints, since a gap clearly above the few percent reported, a non-monotone cost sequence, or an infeasible early iterate would contradict the central claims.

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Extended reading notes

Core claim

The paper's central claim is that the minimum-time problem can be solved by biconvex alternation in a way that is complete and anytime, even though the full problem is nonconvex. The key fact is that fixing either the transition points $q(t_i)$ or the transition velocities $\dot q(t_i)$ makes the remaining problem convex, up to a conservative linearization of the acceleration constraint that underestimates $T_i^2$ (or $1/S_i$) at the current traversal times. Because each convex subproblem is a restriction rather than an approximation that can leave feasibility, SCS can take full steps toward their optima without a line search or trust region; the duration decreases monotonically and any intermediate iterate is feasible. The authors report worst-case duration gaps of about 3.2% against nonconvex solvers on their test battery, with typical gaps near 0.1–1.2% when the stopping tolerance is tightened, and they show that the same package-transfer task completes 28% faster than the trust-region baseline and 50% faster than a waypoint planner.

Load-bearing premise

The whole guarantee rests on every traversal time $T_i$ being strictly positive; Assumption 1 ensures this geometrically, and when an optimal trajectory merely grazes one set, the $S_i=1/T_i$ transformation used in the fixed-transition-point subproblem breaks down.

Editorial extensions

If this is right

  • SCS can be used as an online motion planner for robots operating in convex corridors: planning completes in milliseconds and any interruption yields a usable trajectory.
  • Because no line-search or trust-region parameters are involved, deploying the method in production avoids algorithm-specific tuning; only the Bézier degree and stopping tolerance remain.
  • On the reported benchmark family, SCS scales nearly linearly with the number of safe sets, facets, space dimension, and trajectory degree, so the method should remain practical for large corridors and high-dimensional arms.
  • The small duration gap to nonconvex optimizers (0.1–3.2%) means SCS is appropriate where near-optimal, reliable, fast plans matter more than certified global optimality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same convex-restriction pattern should carry over to fixed-final-time costs that penalize velocity and acceleration, as the paper hints; the reciprocal-time change of variables is not specific to minimizing total time.
  • A natural test of the completeness-versus-optimality distinction is to benchmark SCS against a certified global method in low dimensions; the paper only compares against local nonconvex solvers and heuristic baselines.
  • Because only convex geometry and convex derivative constraints are used, SCS should compose with any convex-corridor generator, including GPU-inflated sets; the package-transfer experiments suggest but do not establish this at scale.
  • The method could also be extended to jerk or higher-order derivative constraints by raising the Bézier degree, but convexity of the resulting constraints would need to be rechecked.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper proposes SCS, an alternating convex-optimization algorithm for the minimum-time motion-planning problem of traversing an ordered sequence of convex sets while respecting convex velocity and acceleration constraints. The authors reformulate the nonconvex infinite-dimensional problem using per-set time scaling, derive two convex subproblems by fixing either transition velocities or transition points and underestimating the time-dependent terms with tangent lines, and initialize the method with a polygonal trajectory. The paper claims monotone convergence, anytime feasibility, no line-search or trust-region parameters, and empirically reports faster runtimes than SNOPT and IPOPT and lower-duration trajectories than trust-region and waypoint baselines, including a two-robot package-transfer benchmark.

Significance. The manuscript is a solid, clearly written contribution to a practical problem. Its strengths include a transparent derivation from elementary convex analysis, where every iterate is feasible by construction and no step-size tuning is required; a fully specified finite-dimensional Bezier implementation in Appendix A; reproducible code; and systematic experiments across problem size, facet count, dimension, and polynomial degree. If the claims hold, the method offers a parameter-light, anytime alternative to nonconvex trajectory optimization within its scope. The principal caveat is that the formal guarantees rely on Assumption 1, which rules out zero traversal times; this is acknowledged in the body but not in the abstract.

major comments (2)
  1. [§II-B, §IX, abstract] The completeness and anytime guarantees are proven only under Assumption 1, which ensures strictly positive traversal times T_i. The reformulation in (15) uses S_i = 1/T_i, so it is not well posed when some optimal traversal time is zero. The abstract states 'guaranteed to converge, returns a feasible trajectory even if stopped early' without this qualification, and the limitations paragraph in §IX proposes an artificial lower bound on traversal times without a proof that the modified method remains complete or anytime, or a bound on the induced suboptimality. I recommend qualifying the abstract (e.g., 'under a mild nondegeneracy assumption') and either analyzing the lower-bound variant or explicitly labeling it as a heuristic.
  2. [§VIII-A] The statement 'SCS is guaranteed to converge monotonically' is used to support the abstract's 'guaranteed to converge.' The proof in §VIII-A shows only that the objective values are nonincreasing and bounded below; it does not establish convergence of the trajectory iterates to a limit or to a stationary point. Since the stopping criterion is a relative-decrease tolerance, the sense of 'converge' should be stated precisely to avoid overstating the theoretical result.
minor comments (4)
  1. [§X-B] The phrases 'runtimes increase by 3060', 'grows by 210', 'by 17.6', and 'by 9.9' are ambiguous; they should be written as 'by a factor of 30–60', 'by a factor of 2–10', etc.
  2. [§V] The variable r_i is used in (12) but defined only indirectly as ˙r_i = ˙q_i/T_i; please state explicitly that r_i = q_i/T_i before (12).
  3. [§VIII-A] The anytime claim should specify that feasibility is guaranteed at iteration boundaries, not during the solve of a convex subproblem, since stopping mid-solve does not necessarily provide a feasible trajectory.
  4. [§II-B] The phrase 'forces our trajectory to cover a nonzero distance within each safe set' is imprecise; Assumption 1 actually guarantees a positive traversal time, which is the property used in the change of variables.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the convex restrictions, initialization, and monotone-improvement argument are derived from explicit inequalities and an independently constructed feasible trajectory; self-citations are contextual or baseline only.

full rationale

The paper's central derivation chain is self-contained. The nonconvex reformulation (7) is obtained by the algebraic change of variables T_i and q_i, with velocity and acceleration expressed as dot(q_i)/T_i and ddot(q_i)/T_i^2. The two convex subproblems are genuine convex restrictions: equation (10) replaces T_i^2 A with the smaller set bar(T_i)(2T_i - bar(T_i))A, justified by the inequality T_i^2 >= bar(T_i)(2T_i - bar(T_i)) and by 0 in A, so any solution of (11) is feasible for (7); similarly, (14) replaces (1/S_i)A with bar(T_i)(2 - bar(T_i)S_i)A, using 1/S_i >= bar(T_i)(2 - bar(T_i)S_i) under 2 >= bar(T_i)S_i. Feasibility of the initialization is proved in Proposition 1 by an explicit polygonal construction, and the relaxation T_i >= 1/S_i in (17) is shown to be lossless by an explicit rescaling. Completeness and the anytime property follow directly from these facts: the initialization produces a feasible trajectory and the convex restrictions never leave the feasible set, so every iterate is feasible. The self-citations to [28] appear as a baseline, as a reproducibility note for Proposition 1, and for standard Bezier-curve facts, none of which is load-bearing for the method's guarantees. The comparison results against SNOPT and IPOPT are external benchmarks, not fitted predictions. The only significant caveat, Assumption 1 and the treatment of zero traversal times, is explicitly acknowledged in Section IX as a limitation rather than being disguised as a result; it does not make the derivation circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The method has no fitted parameters; the only user-specified scalars are the Bézier degree K and tolerance ε, which are not fitted to data. The axioms are standard domain assumptions on the problem data plus the convex-analysis tools used to derive the convex restrictions.

assumptions (5)
  • domain assumption Each safe set Qi is closed, convex, and intersects the next (Qi ∩ Q_{i+1} ≠ ∅).
    Stated in §II. This is required for the feasibility of problem (1), for the initialization convex program (16), and for the polygonal-trajectory construction in Proposition 1.
  • domain assumption The velocity set V and acceleration set A are closed, convex, and contain the origin in their interior.
    Stated in §II. The origin-in-interior condition is needed so that small traversal times make the polygonal initialization feasible, and convexity is used in the convex restrictions of (5b) and (5c) via the discussion in §I-B.
  • domain assumption Assumption 1: qinit ∉ Q2, qterm ∉ Q_{I-1}, and Qi ∩ Q_{i+1} ∩ Q_{i+2} = ∅ for i ≤ I-2.
    Stated in §II-B. Ensures strictly positive traversal times Ti > 0, so the reciprocal variables Si = 1/Ti are finite in the fixed-transition-points subproblem (15).
  • standard math For a convex set S, the set {(x,λ): λ ≥ 0, x ∈ λS} is convex, and the tangent linearization of a convex function is a global underestimator.
    Used to construct the convex restrictions (8)-(10) and (13)-(14); cited from [3, §2.3.3] and standard convex analysis.
  • domain assumption Bézier curves of degree K ≥ 3 can represent straight-line segments with zero endpoint velocity.
    Used in §VIII-A to guarantee the initialization step succeeds in the finite-dimensional implementation; holds by choosing control points p0,p0,p1,p1.

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Cite this review

Pith. "Pith review of A biconvex method for minimum-time motion planning through sequences of convex sets." pith.science (2026). https://pith.science/paper/UQ5B65WR

@misc{pith2026250418978,
  author       = {Pith},
  title        = {Pith review of: A biconvex method for minimum-time motion planning through sequences of convex sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UQ5B65WR}},
  note         = {Machine review of arXiv:2504.18978}
}
read the original abstract

We consider the problem of designing a smooth trajectory that traverses a sequence of convex sets in minimum time, while satisfying given velocity and acceleration constraints. This problem is naturally formulated as a nonconvex program. To solve it, we propose a biconvex method that quickly produces an initial trajectory and iteratively refines it by solving two convex subproblems in alternation. This method is guaranteed to converge, returns a feasible trajectory even if stopped early, and does not require the selection of any line-search or trust-region parameter. Exhaustive experiments show that our method finds high-quality trajectories in a fraction of the time of state-of-the-art solvers for nonconvex optimization. In addition, it achieves runtimes comparable to industry-standard waypoint-based motion planners, while consistently designing lower-duration trajectories than existing optimization-based planners.

Figures

Figures reproduced from arXiv: 2504.18978 by the authors.

Figure 1
Figure 1. Sparrow robot sorting products into bins in the Amazon warehouses. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Example of the motion-planning problem. The safe convex sets to be [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Steps of SCS during the solution of the problem in Fig. 2. In the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: A two-dimensional Bezier curve with control points ´ γ0, . . . , γ4. The area shaded in yellow is the convex hull of the control points. VII. NUMERICAL IMPLEMENTATION The numerical implementation of our method requires a finite-dimensional trajectory parameterization. …
Figure 5
Figure 5. Figure 5: Benchmark problem with I = 5 safe sets in n = 2 dimensions, with m = 4 facets each. The optimal trajectory is shown in blue. runtimes significantly. Conversely, if we decrease the SCS tolerance to, e.g., ε = 10−4 , the objective gap between SCS and the nonconvex solver…
Figure 6
Figure 6. Figure 6: Comparison of SCS with the solvers SNOPT and IPOPT. The runtimes of the three methods are analyzed as functions of multiple problem data. Missing markers correspond to solver failures. The runtimes of SCS grow almost linearly in each experiment (note that the horizonta…
Figure 7
Figure 7. Figure 7: Sparrow robots that move packages between bins in minimum time. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.