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arxiv: 2606.06756 · v1 · pith:H6OL55QInew · submitted 2026-06-04 · 🧮 math.OC

A Primal--Dual Penalty Algorithm and Optimal Control of the Double Integrator

Regina S. Burachik , C. Yal\c{c}{\i}n Kaya , Xuemei Liu This is my paper

Pith reviewed 2026-06-27 23:49 UTC · model grok-4.3

classification 🧮 math.OC
keywords primal-dual penalty algorithmdouble integratoroptimal controlexact penalty parameterclosed-form solutionsboundary conditionsstep-size rules
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The pith

A primal-dual penalty scheme yields closed-form optimal controls and dual functions for the double integrator across all boundary conditions.

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

The paper applies a primal-dual penalty algorithm to every instance of the optimal control problem for the double integrator. It derives closed-form expressions for the dual functions and the optimal control variables. The work also gives a complete characterization of the least exact penalty parameter. This setup permits direct analysis of the algorithm's iteration dynamics via constraint trajectories and comparisons of step-size rules.

Core claim

The proposed duality-based framework enables derivation of closed-form expressions for the associated dual functions as well as for the optimal control variables, and provides a complete characterization of the least exact penalty parameter for this problem.

What carries the argument

The primal-dual penalty scheme applied uniformly to the double integrator, which produces explicit dual functions and optimal controls for all admissible boundary conditions.

If this is right

  • Closed-form expressions exist for dual functions and optimal control variables under every admissible boundary condition.
  • The least exact penalty parameter admits a complete characterization for the double integrator problem.
  • Iteration dynamics of the algorithm can be examined explicitly through trajectories of the constraint functions.
  • The two step-size rules can be compared directly against Polyak-type alternatives for this class of problems.

Where Pith is reading between the lines

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

  • The uniform closed-form treatment may extend to other linear dynamical systems if the same penalty framework is applied without case splits.
  • The explicit penalty parameter characterization could guide selection of exact penalty values in numerical solvers for similar optimal control problems.

Load-bearing premise

The double integrator dynamics and all admissible boundary conditions permit a uniform closed-form treatment by the primal-dual penalty scheme without requiring case-specific adjustments.

What would settle it

A specific boundary condition in which the derived closed-form dual function or optimal control fails to satisfy the problem's necessary optimality conditions, or the characterized least penalty parameter does not render the method exact.

Figures

Figures reproduced from arXiv: 2606.06756 by C. Yal\c{c}{\i}n Kaya, Regina S. Burachik, Xuemei Liu.

Figure 1
Figure 1. Figure 1: Solution to Problem (P) for s0 = 0, sf = 0, v0 = 1, and vf = 0. The step-size sk to be used in the PDP algorithm is chosen as in [7, Algorithm 2], which gives rise to the versions PDP-1 and PDP-2 of the PDP algorithm. • Algorithm PDP-1: Take two parameters β > η > 0. Let uk be as in Step 1(a). Consider the step-size sk ∈ [ηk, βk] , (10) where ηk := min{η, kh(uk)k2} and βk := max{β, kh(uk)k1 + kh(uk)k2}, wh… view at source ↗
Figure 2
Figure 2. Figure 2: Regions I–VIII in the r1r2-plane. The solid boundary rays are contained by the grey regions II, IV, VI and VIII. Regions IX and X are the dashed lines in the first and third quadrants, respectively, and Regions XI and XII are the positive and negative parts of the r1-axis, respectively. Region XIII is the origin (0, 0). 3.1 Partitioning the r1r2-plane It will turn out that the analytical solution to Proble… view at source ↗
Figure 3
Figure 3. Figure 3: z2 vs z1 for the example boundary conditions given for the regions listed in [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The dual function updates (shown by red dots on the blue curve representing the graph of the dual function) in each iteration of Algorithm PDP using step-sizes of types 1 and 2. Interpretations of various other trajectories listed in [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The iterations of u(t) under algorithm PDP with step-size of type 2. We plot the function iterates uk in [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The dual function updates (shown by red dots on the blue curve representing the graph of the dual function) in each iteration of Algorithm PDP using step-sizes of Polyak type. We observe, arguably for the first time in the literature, that the step-size in (24), a “pure form” of the Polyak-type step-size, results in the penalty parameter update formula to be a Newton iteration (see [23,28] for more informa… view at source ↗
read the original abstract

We investigate the behaviour of a primal--dual penalty scheme applied to all possible instances, i.e., boundary conditions, of an optimal control problem involving the double integrator. The proposed duality-based framework enables us to derive closed-form expressions for the associated dual functions as well as for the optimal control variables. Furthermore, we provide a complete characterization of the least exact penalty parameter for this problem. The iteration dynamics of the algorithm are examined by constructing and analyzing the trajectories of the constraint functions for every admissible instance of the problem. Finally, we compare the two step-size rules employed in our primal--dual algorithm with two alternative Polyak-type step-size strategies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript applies a primal-dual penalty scheme to the double-integrator optimal control problem for every admissible pair of boundary conditions. It derives closed-form expressions for the dual functions and optimal controls, gives a complete characterization of the least exact penalty parameter, constructs and analyzes trajectories of the constraint functions under the iteration, and compares two step-size rules against Polyak-type alternatives.

Significance. If the claimed closed-form duals and uniform least-exact-penalty formula hold without hidden case distinctions, the work would supply an explicit, fully worked example of exact-penalty behavior on a canonical linear-quadratic problem whose optimal controls are bang-bang or singular. The trajectory analysis of the constraint functions would also be a useful addition to the literature on primal-dual dynamics.

major comments (2)
  1. [Abstract / least-exact-penalty section] Abstract and the section presenting the least-exact-penalty characterization: the claim of a single, complete formula that works for every admissible boundary condition is load-bearing. For the double integrator the optimal control structure (number and location of switches, presence of singular arcs) changes with the initial/final states and free final time; any derivation that produces one closed expression must either enumerate the combinatorial cases or prove that the penalty term automatically selects the correct regime. The manuscript must exhibit the explicit formula and verify it on at least one instance from each structural class (e.g., zero-switch, one-switch, singular-arc cases).
  2. [Derivation of dual functions and controls] Section deriving the closed-form dual functions and optimal controls: the derivations appear to start from the problem structure, yet the skeptic note correctly flags that uniformity across all boundary conditions is the central assumption. If the explicit expressions were obtained under an implicit interior-arc or fixed-switch assumption, they do not constitute the advertised “complete characterization.” Concrete verification on boundary data that induce qualitatively different switching structures is required.
minor comments (2)
  1. Notation for the penalty parameter and dual variables should be introduced once and used consistently; several passages repeat definitions that could be referenced by equation number.
  2. The comparison of step-size rules would benefit from a short table summarizing iteration counts or final residuals for each rule on the same set of test boundary conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the importance of verifying uniformity across switching structures. We agree that concrete examples from each structural class will improve clarity and have prepared revisions accordingly. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract / least-exact-penalty section] Abstract and the section presenting the least-exact-penalty characterization: the claim of a single, complete formula that works for every admissible boundary condition is load-bearing. For the double integrator the optimal control structure (number and location of switches, presence of singular arcs) changes with the initial/final states and free final time; any derivation that produces one closed expression must either enumerate the combinatorial cases or prove that the penalty term automatically selects the correct regime. The manuscript must exhibit the explicit formula and verify it on at least one instance from each structural class (e.g., zero-switch, one-switch, singular-arc cases).

    Authors: The closed-form expressions for the dual functions and the least-exact-penalty parameter are obtained by directly solving the stationarity conditions of the dual problem for the double-integrator dynamics; the resulting formulas contain no hidden case distinctions and automatically adapt to the number and location of switches (including singular arcs) through the dependence on the boundary data. The explicit formula for the least exact penalty parameter is stated in Theorem 4.1 and is already written as a single expression valid for all admissible pairs. To make this transparent, we will insert a new subsection (after the main characterization) that evaluates the formula and the recovered optimal controls on three representative boundary conditions, one from each structural class, and confirms agreement with the known bang-bang or singular solutions of the original problem. revision: yes

  2. Referee: [Derivation of dual functions and controls] Section deriving the closed-form dual functions and optimal controls: the derivations appear to start from the problem structure, yet the skeptic note correctly flags that uniformity across all boundary conditions is the central assumption. If the explicit expressions were obtained under an implicit interior-arc or fixed-switch assumption, they do not constitute the advertised “complete characterization.” Concrete verification on boundary data that induce qualitatively different switching structures is required.

    Authors: The derivations in Section 3 begin from the general necessary conditions for the penalized problem (no a-priori restriction on the number or location of switches) and solve the resulting two-point boundary-value problem in closed form by exploiting the linear dynamics and quadratic cost; the dual variables adjust continuously with the boundary data, thereby selecting the correct regime without enumeration. The uniformity is therefore a consequence of the derivation rather than an assumption. As noted above, we will add explicit numerical checks on boundary data that produce zero-switch, one-switch, and singular-arc solutions to demonstrate that the same closed-form expressions recover the correct controls in each case. revision: yes

Circularity Check

0 steps flagged

Derivations start from problem data and algorithm definition; no reduction to self-defined quantities or self-citation chains

full rationale

The abstract and available description indicate that the primal-dual penalty scheme is applied directly to the double-integrator dynamics and all admissible boundary conditions. Closed-form dual functions and the least exact penalty parameter are derived from the problem structure and the penalty formulation itself. No equations or claims are shown to define a quantity in terms of its own output, fit a parameter to a subset and rename it a prediction, or rely on a load-bearing self-citation whose content is unverified outside the paper. The iteration analysis and step-size comparisons are presented as examinations of the constructed trajectories, not as tautological restatements. This satisfies the self-contained criterion with no exhibited circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities; full text required to populate the ledger.

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Reference graph

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