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REVIEW 2 major objections 6 minor 14 references

Reformulating powered-descent guidance into a strongly convex program with exact state elimination yields a fixed-budget solver that averages 68.2 microseconds per solve with a 0.033% fuel penalty.

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2026-08-01 02:55 UTC pith:4WJC4M25

load-bearing objection A solid, honestly-scoped engineering paper: exact condensation plus strong convexity buys a 93-iteration/68 µs solve for one Mars landing case; the math is clean, and the main limitation is the single-scenario tuning the authors themselves acknowledge. the 2 major comments →

arxiv 2607.25260 v1 pith:4WJC4M25 submitted 2026-07-28 math.OC

Microsecond-Class Powered-Descent Optimization via Exact Condensation and Strong Convex Regularization

Wenbo Li , Ziqi Xu , Dai Shen , Shengping Gong This is my paper
classification math.OC MSC 90C25
keywords powered descent guidanceconvex optimizationstrong convexitystate condensationproximal operatorfirst-order methodsreal-time guidancefuel-optimal control
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that the computational core of fuel-dominant powered-descent guidance can be made microsecond-fast through formulation design rather than faster hardware or a new convexification theorem. It adds a small, dimensionally consistent energy term to the fuel objective, giving the problem explicit strong convexity and a unique optimal control. A terminal-sensitivity recursion then eliminates every intermediate state exactly, leaving only six terminal equality constraints and 90 control variables. The resulting problem has a closed-form proximal operator (group shrinkage plus magnitude clipping) and is solved with a fixed 93-iteration first-order loop in fixed-size C arrays. On a Mars landing test case, the solver reaches the reference tolerance in 68 microseconds mean (P99 128 microseconds), while the fuel metric rises only 0.033% above the pure-fuel optimum.

Core claim

The central claim is that a terminal-constrained, convex-ball thrust powered-descent problem can be condensed into a strongly convex program whose online solution fits in a few kilobytes and runs in tens of microseconds. Starting from a full-state epigraph formulation with 300 primal variables and 174 dynamics multipliers, exact condensation via a terminal-sensitivity recursion yields a problem with 90 control variables and only six terminal multipliers. A low-weight energy term (weight λ=0.02) provides a strong-convexity modulus of λ/2, guaranteeing a unique control and accelerating first-order convergence. The fuel norm and thrust ball share radial symmetry, giving an exact proximal map th

What carries the argument

Three structural reductions carry the argument. First, strong convex regularization: a dimensionlessly scaled energy term λ·(Δt/t_f u_max^2)·Σ w_k||u_k||^2 is added to the fuel metric, making the objective μ-strongly convex with μ=λ/2 and a unique minimizer. Second, exact dynamics condensation: a terminal-sensitivity recursion produces a 6×3N matrix G (row-normalized to H_c) that maps the stacked control vector directly to the terminal state, eliminating all intermediate states and interval multipliers; the condensed problem has 90 primal variables and six terminal multipliers. Third, a closed-form radial proximal operator: because both the weighted fuel norm and the spherical thrust bound d

Load-bearing premise

The headline numbers (93 iterations, 68.2 µs, 0.033% penalty) come from one fixed Mars-landing scenario with stepsizes and extrapolation tuned offline to that scenario; if the initial state, target, time grid, or dynamics change, the fixed 93-iteration budget may no longer reach the 10^-3 reference tolerance without retuning.

What would settle it

Run the fixed-budget (93-iteration) implementation on a different powered-descent instance of the same model—same thrust limit and grid but random initial states drawn from a reasonable envelope—and compare the relative reference error at iteration 93 against 10^-3; if a non-negligible fraction of instances exceed 10^-3, the claim that the fixed budget suffices for the stated class is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The online kernel uses only two 6×90 matrix-vector products and thirty independent 3-D proximal maps per iteration, so compute and memory per iteration scale linearly in the number of control nodes with very small constants.
  • Strong convexity plus a fixed 93-iteration budget makes the execution path deterministic: no factorizations, line searches, or data-dependent early-exit branches, which is favorable for certification and worst-case execution-time analysis.
  • The fuel penalty at λ=0.02 is 0.033%, and increasing λ trades fuel for speed (e.g., 18 iterations at λ=1 with a ~5.3% fuel increase), so the curvature parameter provides a tunable fuel-versus-latency frontier.
  • Condensation is exact for the stated class: any feasible condensed control maps back to a feasible full-state trajectory via one forward recursion, with no loss in optimality of the control-only objective.
  • The approach is tailored to translational landing with terminal constraints, zero minimum thrust, and no path constraints; extending to 6-DoF, nonzero lower thrust, or path constraints requires a new proximal decomposition or use inside a successive-convexification loop.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If embedded in a successive-convexification loop for nonlinear 6-DoF guidance, the microsecond kernel would enable dozens of convex subproblem solves per guidance cycle, plausibly bringing real-time 6-DoF landing guidance to flight hardware—an extension the paper names as future work.
  • The strong-convexity modulus μ=λ/2 is a natural online tuning knob: a guidance computer could adjust λ in flight based on available compute budget or emergency state, trading a few percent of fuel for guaranteed convergence speed.
  • The condensation method may transfer to other optimal-control problems with linear dynamics, terminal-only constraints, and control-only path constraints, such as rendezvous, docking, or small-body landing, whenever the terminal dimension is small.
  • The reported timing is on a desktop x86 processor; the same structure could map to embedded vector hardware, but cache behavior, fixed-point arithmetic, and worst-case interrupt timing would need independent validation.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper develops a formulation-to-code pipeline for terminal-constrained powered descent. Three structural reductions are combined: (i) a dimensionally consistent energy–fuel objective that adds strong convexity with modulus μ=λ/2; (ii) exact elimination of intermediate states and interval dynamics multipliers via a terminal-sensitivity recursion, reducing the N=30 baseline from 300 primal variables and 174 equality multipliers to 90 controls and 6 terminal multipliers; and (iii) an exact closed-form proximal operator for the radial fuel norm plus thrust ball, avoiding epigraph variables and inner iterations. The condensed problem is solved with a fixed-budget extrapolated proportional–integral projected-gradient (xPIPG) method implemented in fixed-size C17 arrays. For a single Mars landing test case with λ=0.02, the method reaches a relative reference-solution error of 10^-3 in 93 iterations (vs. 2744 for the pure-fuel epigraph baseline), with a fuel-metric penalty of 0.033%; mean end-to-end solve time is 68.2 µs (P99 = 128.1 µs) on an Intel i7-10875H. The paper states that the thrust set is already a convex ball and that the contribution is fast solution of the convex core, not a new lossless-convexification theorem.

Significance. If the results hold, the paper demonstrates that formulation design—strong convexity, exact condensation, and a closed-form proximal map—can shrink the convex guidance core to microsecond-scale, fixed-memory computation. The theoretical reductions (Propositions 1–3) are correct and clean, and the cross-language agreement between the C17 and MATLAB implementations (≈1e-12) plus the careful reporting of percentiles are notable strengths. However, the headline quantitative claims (93 iterations, 68.2 µs, P99 128.1 µs) are point estimates for one fixed test case, with hyperparameters (ω, ρ, c, J_max) tuned to that case. The paper openly acknowledges this limitation, but the title and abstract generalize the result; the contribution is best understood as a proof-of-concept architecture rather than a general performance guarantee. The lack of an ablation also means the iteration reduction is not attributable to any single mechanism, though that is explicitly stated in the text.

major comments (2)
  1. [§V.B–V.C, Table 2] The central quantitative claims—93 iterations, 68.2 µs mean, P99 128.1 µs, 0.033% fuel penalty—are established for a single fixed test case. §V.B states that ω=0.15, ρ=1.995, and c=0.99 were 'selected by an offline candidate search for the fixed test case,' and §V.C sets J_max=93 as the first iteration at which e_ref reaches 10^-3 for that case. For a different initial state, target, grid, or dynamics, H_c changes and the spectral stepsizes and fixed budget are not re-derived; no sensitivity, Monte Carlo, or worst-case analysis is reported. The paper acknowledges this limitation (§V.C, §VIII.C), but the title and abstract present 'microsecond-class powered-descent optimization' as a general capability. This is load-bearing: if 93 iterations fails on nearby operating points, the headline latency is a single-scenario artifact. Please either (a) add a robustness study over an explicit opera
  2. [§VII.A, Eq. (43)] The weighted reference solution is generated using 'a high-accuracy condensed primal–dual warm start followed by sequential quadratic programming with 10^-10 constraint and optimality tolerances.' Because the reference is used to compute the relative error metric e_ref, the convergence claim (93 iterations to 10^-3) is measured against a reference initialized from the same solver's output. This creates a circularity risk: if the SQP polish is not fully converged to the true optimum, or if the warm start biases the reference, the reported iteration count could be optimistic. The SQP tolerances suggest high accuracy, but the independence of the reference should be demonstrated by a second solver (e.g., coneprog or CVX on the condensed problem) or by explicitly reporting the final constraint and optimality residuals of the reference. Please add this verification or explain why the warm-star
minor comments (6)
  1. [§VII.C, §VIII.A] The factor-29.5 iteration reduction is attributed to three mechanisms acting together, and the paper correctly states that no single-factor ablation is performed. However, the abstract's phrasing 'iteration count decreases from 2744 for the pure-fuel full-state epigraph baseline to 93' could be read as a direct comparison. Consider adding a sentence in the abstract or conclusions clarifying that this is a combined effect, not a causal attribution.
  2. [§V.C, Eqs. (44)–(45)] Implementable residuals r_eq and r_fp are defined but never used in the numerical results. A figure showing these residuals versus iteration alongside e_ref would strengthen the practical utility of the fixed-budget method, since e_ref is unavailable onboard.
  3. [§VII.B, Eqs. (50)–(51)] The terminal errors after 93 iterations include a y-position error of about 0.64 m. The paper notes these are not mission-level tolerances. Please state the implied physical terminal tolerance corresponding to e_ref≤10^-3 (e.g., in meters and m/s) to help readers judge whether the method meets typical powered-descent requirements.
  4. [§II.C, Eq. (10)] The objective is called 'dimensionless.' Please confirm that each term in Eq. (10) is dimensionless and state the dimensions explicitly in a sentence or footnote; the current text is brief.
  5. [§VII.A] The scalar scale matrices P_r, P_v, P_u (0.943263, 0.843542, 1.256780) are introduced without rationale. Since stepsizes and iteration counts depend on scaling, please state how these were selected and whether the 93-iteration budget is stable to nearby scaling choices.
  6. [§VI.B] The 'no dynamic memory' property is a strength. Please also report the processor frequency at which timing was performed (the i7-10875H has a wide turbo range) and note whether the Windows scheduling outliers (~1.2 ms) were excluded or included in the P99 statistic.

Circularity Check

0 steps flagged

No significant circularity: the mathematical reductions are self-contained; the fixed-budget iteration count and timing are measured, explicitly calibrated, and not presented as free predictions.

full rationale

The paper's central derivation chain is self-contained. Propositions 1–3 are proven directly: Eqs. (20)–(23) show that terminal condensation is an exact elimination under an invertible row scaling; Proposition 2 follows from P ⪰ (λ/2)I and standard convex analysis [13]; Proposition 3 reduces the radial proximal problem to a scalar projection on [0, ū] with no dependence on fitted quantities. The calibrated elements are the algorithmic hyperparameters ω, ρ, c, and J_max. Section V.B states they were "selected by an offline candidate search for the fixed test case," and Section V.C defines J_max=93 as the first iteration at which e_ref reaches 10^-3 for that case. The paper does not present 93 as a theorem, worst-case bound, or Monte Carlo quantile; it explicitly limits the fixed budget to the "offline-tested operating envelope" and lists Monte Carlo coverage among the open items for flight qualification (Section VIII.C). The reported 68.2 µs is the measured wall-clock time to execute this calibrated 93-iteration kernel, not a predicted latency derived from the inputs. Thus the calibrated iteration budget affects external validity, not the internal derivation, and no equation is equivalent to its own input by construction. There is also no load-bearing self-citation: the cited primal–dual, proximal, and convex-analysis references are standard or by non-overlapping authors, and no uniqueness theorem is imported from the present authors' prior work.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

The paper's central claim relies on a handful of explicit modeling restrictions: the thrust set is a convex ball with zero minimum thrust; no state-path constraints exist; mass is fixed; and the discretized trapezoidal model is the problem actually solved. These are honestly stated. The numerical performance depends on several hand-tuned quantities (energy weight, stepsize ratio, extrapolation cache, safety factor, scale matrices, fixed iteration budget), which are calibrated to the single test case rather than predicted from theory.

free parameters (6)
  • energy weight lambda = 0.02
    Chosen to balance fuel-dominance against strong convexity; the central 93-iteration result and 0.033% fuel penalty are measured at this value.
  • primal-dual ratio omega = 0.15 (for lambda=0.02)
    Offline candidate search for the fixed test case; directly controls the stepsize alpha and convergence rate.
  • extrapolation parameter rho = 1.995
    Offline-tuned extrapolation weight in the xPIPG iteration; affects convergence behavior.
  • safety factor c = 0.99
    Chosen to keep the spectral stepsize condition below 1; yields alpha=1.843372658 and beta=0.276505899.
  • iteration budget J_max = 93
    Fixed from the offline reference-error test for the test case; the timing implementation runs exactly 93 iterations.
  • scale matrices P_r, P_v, P_u = 0.943263, 0.843542, 1.256780 (scalar multiples of I3)
    Hand-selected conditioning scalings applied before condensation; affect terminal row balance and spectral stepsize.
axioms (5)
  • domain assumption The trapezoidal discretization is the problem solved; continuous-time feasibility/optimality is not certified.
    Eqs. (4)-(5) define the dynamics used everywhere; no integration-error bound to the continuous ODE is provided.
  • domain assumption Thrust envelope is a convex ball 0 ≤ ||u|| ≤ u_max (no nonzero lower bound, no pointing cone).
    Section II.A and the abstract explicitly state this scoping; it is what makes the proximal map and full condensation exact.
  • domain assumption No state-path constraints (glide-slope, obstacles, state-triggered constraints) are present.
    Section III.C states full condensation is exact only without such constraints.
  • domain assumption The SQP-generated weighted reference is a faithful near-optimal solution for the condensed problem.
    Section VII.A: the weighted reference uses 'a high-accuracy condensed primal–dual warm start followed by sequential quadratic programming with 10^-10 tolerances'; the reported iteration counts and fuel penalty are measured against this reference.
  • standard math Cauchy–Schwarz inequality and standard convex-analysis facts used in the proofs of Propositions 1–3.
    Standard background; uncontested.

pith-pipeline@v1.3.0-alltime-deepseek · 10913 in / 16596 out tokens · 149304 ms · 2026-08-01T02:55:47.229746+00:00 · methodology

0 comments
read the original abstract

Fuel-dominant powered descent can be written as a convex program, but the usual full-state epigraph formulation still carries many state variables, fuel epigraph variables, and dynamics equalities. In addition, the pure-fuel objective provides no strong-convexity curvature. This paper combines three structural reductions. First, a dimensionally consistent low-weight energy term makes the control solution unique. Second, a terminal-state sensitivity recursion eliminates every intermediate state exactly; invertible row normalization turns the 30-node baseline with 300 primal variables and 174 equality multipliers into a problem with 90 control variables and six terminal multipliers. Third, the shared radial structure of the fuel norm and thrust ball gives an exact closed-form proximal operator consisting of group shrinkage followed by magnitude clipping. The condensed problem is solved with a fixed-budget extrapolated proportional--integral projected-gradient iteration implemented in fixed-size C17 arrays. In a Mars powered-descent case with energy weight 0.02 and relative reference-solution tolerance $10^{-3}$, the iteration count decreases from 2744 for the pure-fuel full-state epigraph baseline to 93, while the fuel metric increases by only 0.033\%. The mean end-to-end solve time is \SI{68.2}{\micro\second}, and P99 is \SI{128.1}{\micro\second}, on an Intel i7-10875H. Because the thrust set in this test case is already a convex ball, the contribution is fast solution of the convex core rather than a new lossless-convexification theorem.

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references · 5 canonical work pages

  1. [1]

    Convex Programming Approach to Powered Descent Guidance for Mars Landing,

    Açıkmeşe, B., and Ploen, S. R., “Convex Programming Approach to Powered Descent Guidance for Mars Landing,”Journal of Guidance, Control, and Dynamics, Vol. 30, No. 5, 2007, pp. 1353–1366. doi:10.2514/1.27553

    doi:10.2514/1.27553 2007

  2. [2]

    Minimum-Landing-ErrorPowered-DescentGuidanceforMarsLandingUsing ConvexOptimization,

    Blackmore, L., Açıkmeşe, B., andScharf, D.P., “Minimum-Landing-ErrorPowered-DescentGuidanceforMarsLandingUsing ConvexOptimization,”JournalofGuidance,Control,andDynamics,Vol.33,No.4,2010,pp.1161–1171. doi:10.2514/1.47202

    doi:10.2514/1.47202 2010

  3. [3]

    Successive Convexification for Real-Time Six-Degree-of-Freedom Powered Descent Guidance with State-Triggered Constraints,

    Szmuk, M., Reynolds, T. P., Açıkmeşe, B., Mesbahi, M., and Carson, J. M., “Successive Convexification for Real-Time Six-Degree-of-Freedom Powered Descent Guidance with State-Triggered Constraints,”Journal of Guidance, Control, and Dynamics, Vol. 43, No. 8, 2020, pp. 1399–1413. doi:10.2514/1.G004549

    doi:10.2514/1.g004549 2020

  4. [4]

    Advances in Trajectory Optimization for Space Vehicle Control,

    Malyuta, D., Yu, Y., Elango, P., and Açıkmeşe, B., “Advances in Trajectory Optimization for Space Vehicle Control,”Annual Reviews in Control, Vol. 52, 2021, pp. 282–315. doi:10.1016/j.arcontrol.2021.04.001

    work page doi:10.1016/j.arcontrol.2021.04.001 2021

  5. [5]

    CVXGEN:ACodeGeneratorforEmbeddedConvexOptimization,

    Mattingley,J.,andBoyd,S.,“CVXGEN:ACodeGeneratorforEmbeddedConvexOptimization,”OptimizationandEngineering, Vol. 13, No. 1, 2012, pp. 1–27. doi:10.1007/s11081-011-9176-9

    work page doi:10.1007/s11081-011-9176-9 2012

  6. [6]

    ECOS: An SOCP Solver for Embedded Systems,

    Domahidi, A., Chu, E., and Boyd, S., “ECOS: An SOCP Solver for Embedded Systems,”2013 European Control Conference, 2013, pp. 3071–3076. doi:10.23919/ECC.2013.6669541

    arXiv 2013

  7. [7]

    OSQP:AnOperatorSplittingSolverforQuadraticPrograms,

    Stellato,B.,Banjac,G.,Goulart,P.,Bemporad,A.,andBoyd,S.,“OSQP:AnOperatorSplittingSolverforQuadraticPrograms,” Mathematical Programming Computation, Vol. 12, 2020, pp. 637–672. doi:10.1007/s12532-020-00179-2

    doi:10.1007/s12532-020-00179-2 2020

  8. [8]

    Proportional-Integral Projected Gradient Method for Conic Optimization,

    Yu, Y., Elango, P., Topcu, U., and Açıkmeşe, B., “Proportional-Integral Projected Gradient Method for Conic Optimization,” Automatica, Vol. 142, 2022, p. 110359. doi:10.1016/j.automatica.2022.110359. 18

    arXiv 2022

  9. [9]

    Extrapolated Proportional-Integral Projected Gradient Method for Conic Optimization,

    Yu, Y., Elango, P., Açıkmeşe, B., and Topcu, U., “Extrapolated Proportional-Integral Projected Gradient Method for Conic Optimization,”IEEE Control Systems Letters, Vol. 7, 2023, pp. 73–78. doi:10.1109/LCSYS.2022.3186647

    arXiv 2023

  10. [10]

    A Customized First-Order Solver for Real-Time Powered-Descent Guidance,

    Elango, P., Kamath, A. G., Mesbahi, M., Açıkmeşe, B., Yu, Y., and Carson, J. M., “A Customized First-Order Solver for Real-Time Powered-Descent Guidance,”AIAA SCITECH 2022 Forum, 2022, pp. AIAA 2022–0951. doi:10.2514/6.2022-0951

    work page doi:10.2514/6.2022-0951 2022

  11. [11]

    Customized Real-Time First-Order Methods for Onboard Dual Quaternion-Based 6-DoF Powered-Descent Guidance,

    Kamath, A. G., Elango, P., Kim, T., Mceowen, S., Mesbahi, M., Açıkmeşe, B., Yu, Y., and Carson, J. M., “Customized Real-Time First-Order Methods for Onboard Dual Quaternion-Based 6-DoF Powered-Descent Guidance,”AIAA SCITECH 2023 Forum, 2023, pp. AIAA 2023–2003. doi:10.2514/6.2023-2003

    work page doi:10.2514/6.2023-2003 2023

  12. [12]

    Scaling and Balancing for High-Performance Computation of Optimal Controls,

    Ross, I. M., Gong, Q., Karpenko, M., and Proulx, R. J., “Scaling and Balancing for High-Performance Computation of Optimal Controls,”Journal of Guidance, Control, and Dynamics, Vol. 41, No. 10, 2018, pp. 2086–2097. doi:10.2514/1.G003382

    work page doi:10.2514/1.g003382 2018

  13. [13]

    Boyd, S., and Vandenberghe, L.,Convex Optimization, Cambridge University Press, Cambridge, 2004

    2004

  14. [14]

    Proximal Algorithms,

    Parikh, N., and Boyd, S., “Proximal Algorithms,”Foundations and Trends in Optimization, Vol. 1, No. 3, 2014, pp. 127–239. doi:10.1561/2400000003. 19

    doi:10.1561/2400000003 2014