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REVIEW 2 major objections 1 minor 96 references

Disciplined Nonlinear Programming

T0 review · 2 major / 1 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read DNLP defines composition rules so nonlinear programs mixing smooth and nonsmooth convex or concave functions can be converted automatically into equivalent smooth NLPs.

desk verdict DNLP adds composition rules and a CVXPY extension to let users mix smooth functions with nonsmooth convex/concave ones in NLPs and get automatic lossless canonicalization to standard form. read the letter →

arxiv 2606.02896 v1 pith:HSPSQXNV submitted 2026-06-01 math.OC

Daniel Cederberg , William Zhang , Parth Nobel , Stephen Boyd This is my paper
classification math.OC
keywords disciplinednonlinearprogrammingnonsmoothoptimizationcanonicalizationconvexfunctionsconcavemodelinglanguages
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

The paper introduces disciplined nonlinear programming as a syntax for writing nonlinear programs that freely combine smooth functions with selected nonsmooth convex and concave ones. Composition rules restrict how the nonsmooth pieces may be used, so that the entire problem can be rewritten by a parser into standard NLP form. The rewrite relaxes each nonsmooth function in a way that leaves the set of optimal solutions unchanged. A reader would care because the method lets modelers use convenient nondifferentiable terms without performing manual reformulations or switching to specialized solvers.

What carries the argument

The DNLP composition rules for nonsmooth convex and concave functions, which define the class of problems that admit lossless relaxation into standard NLP form.

What would settle it

An explicit DNLP problem whose canonicalized NLP version yields a different optimal value or different solution set from the original problem.

Watch

Extended reading notes

Core claim

DNLP supplies a set of rules that govern the use of nonsmooth convex and concave functions inside nonlinear programs; under those rules an automatic canonicalization step produces an equivalent smooth NLP whose solutions coincide with those of the original problem.

Load-bearing premise

The stated rules for combining nonsmooth convex or concave functions with smooth ones guarantee that the relaxed problem has exactly the same solutions as the original.

Editorial extensions

If this is right

  • Modelers can include useful nondifferentiable convex and concave terms directly in their problem statements.
  • Existing smooth NLP solvers become applicable to a wider set of problems without manual reformulation.
  • Problem initialization can be performed on the canonicalized form rather than on the original expression.
  • The same parser that checks the composition rules also produces the solver-ready instance.

Reading between the lines

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

  • The method could be applied to problems in which nonsmooth terms appear inside more complex expressions than the current rules allow, once additional composition patterns are certified.
  • Because the canonicalized form is produced automatically, it may reduce transcription errors that occur when users perform the same relaxation by hand.
  • Extending the rule set to cover additional nonsmooth functions would enlarge the modeling language without changing the underlying solver interface.
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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 / 1 minor

Summary. The paper introduces disciplined nonlinear programming (DNLP), a modeling syntax inspired by disciplined convex programming (DCP). DNLP permits free mixing of smooth functions with nonsmooth convex and concave functions under explicit composition rules. The central claim is that DNLP problems can be automatically canonicalized to standard nonlinear programming (NLP) form via lossless relaxation of the nonsmooth functions, enabling direct use with smooth NLP solvers; an open-source CVXPY extension implements the transformation.

Significance. If the composition rules and canonicalization are shown to preserve equivalence, the work would meaningfully extend disciplined modeling techniques from convex to general nonlinear programs, allowing modelers to incorporate useful nondifferentiable convex/concave atoms while retaining solver compatibility and simpler initialization. The open-source implementation is a concrete strength.

major comments (2)
  1. [Abstract] Abstract: the central claim that canonicalization 'relaxes nonsmooth convex and concave functions in a lossless way' and produces an 'equivalent NLP form' is asserted without any derivation of the composition rules, any worked example, or any verification that solutions of the relaxed NLP match those of the original DNLP; this is load-bearing for the automatic-canonicalization guarantee.
  2. [The description of the language] The description of the language (full text): the rules governing composition of nonsmooth convex/concave functions with smooth ones are presented as the mechanism that guarantees equivalence, yet no formal statement or inductive argument is supplied showing that the relaxation step preserves the feasible set and optimal value.
minor comments (1)
  1. The manuscript would benefit from a short table listing the allowed atoms and their curvature/convexity tags, analogous to the DCP atom library.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments on our manuscript. We respond to each major comment below and outline the revisions we will make to address the concerns regarding the justification of the canonicalization process.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that canonicalization 'relaxes nonsmooth convex and concave functions in a lossless way' and produces an 'equivalent NLP form' is asserted without any derivation of the composition rules, any worked example, or any verification that solutions of the relaxed NLP match those of the original DNLP; this is load-bearing for the automatic-canonicalization guarantee.

    Authors: We acknowledge that the abstract is concise and does not include derivations or examples. In the revised manuscript, we will modify the abstract to briefly reference the composition rules and the formal equivalence argument provided in the body of the paper. We will also include a worked example early in the manuscript to demonstrate the lossless relaxation and verify equivalence for a simple case. revision: yes

  2. Referee: [The description of the language] The description of the language (full text): the rules governing composition of nonsmooth convex/concave functions with smooth ones are presented as the mechanism that guarantees equivalence, yet no formal statement or inductive argument is supplied showing that the relaxation step preserves the feasible set and optimal value.

    Authors: The referee correctly notes the absence of a formal inductive argument in the current version. We will add a new subsection in the language description that provides a formal statement of the composition rules and an inductive proof that the canonicalization via relaxation preserves both the feasible set and the optimal value. This will directly address the load-bearing claim of equivalence. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The manuscript defines a new modeling syntax (DNLP) and associated composition rules for mixing smooth functions with nonsmooth convex/concave ones, then describes an explicit canonicalization procedure that produces an equivalent smooth NLP. These rules and the transformation algorithm are presented directly as the paper's contribution rather than derived from prior fitted quantities, self-referential definitions, or load-bearing self-citations. The reference to DCP provides background context for the relaxation technique but does not substitute for the new DNLP rules; the implementation in CVXPY is a software artifact whose correctness is externally verifiable by execution on test problems. No equations or claims reduce to their own inputs by construction.

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

Abstract-only review; no free parameters, axioms, or invented entities are identifiable. The approach rests on standard assumptions from convex analysis and DCP composition rules, but these cannot be audited without the full text.

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Cite this review

Pith. "Pith review of Disciplined Nonlinear Programming." pith.science (2026). https://pith.science/paper/HSPSQXNV

@misc{pith2026260602896,
  author       = {Pith},
  title        = {Pith review of: Disciplined Nonlinear Programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HSPSQXNV}},
  note         = {Machine review of arXiv:2606.02896}
}
read the original abstract

We introduce disciplined nonlinear programming (DNLP), a syntax for specifying nonlinear programming problems. DNLP is inspired by disciplined convex programming (DCP) and allows smooth functions to be freely mixed with nonsmooth convex and concave functions, with rules governing how the nonsmooth functions can be used. Problems expressed in DNLP form can be automatically canonicalized to a standard nonlinear programming (NLP) form and passed to a suitable NLP solver. As in DCP, the canonicalization relaxes nonsmooth convex and concave functions in a lossless way, allowing them to be handled by NLP solvers that require smooth functions. In addition to extending NLP to include useful nondifferentiable convex and concave functions, transforming the original problem to an equivalent NLP form offers several advantages, including simpler problem initialization. We describe the language and our open-source implementation of DNLP as an extension of CVXPY, a parser for DCP.

Figures

Figures reproduced from arXiv: 2606.02896 by the authors.

Figure 1
Figure 1. The red dots denote the initial point and the iterate that Ipopt converged to. (a) Ipopt converges (in two iterations) to a local minimizer that is not the nearest one to the initial point. (b) Ipopt converges (in thirteen iterations) to a saddle point. (c) Ipopt converges (in eight iterations) to an infeasible point. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Expression tree for the L-convex expression abs(quad form(x, A) + c). the definitions given in §3.2. Finally, it verifies that the objective and constraints conform to the DNLP ruleset described in §3.3. After the parser has verified that the problem conforms to DNLP, the canoni￾calizer traverses the expression trees from the root to the leaves and transforms the user-specified problem, distinguishing between how sm… view at source ↗
Figure 3
Figure 3. Shortest path connecting two points while avoiding circular obstacles. Alternative DNLP-compliant formulations. The constraint ∥xi+1 − xi∥ 2 2 ≤ (L/n) 2 can also be expressed as ∥xi+1 − xi∥2 ≤ L/n, which is DNLP-compliant because the left-hand side is L-convex. Since the objective is decreasing in L, these constraints are tight at optimality, so we can also replace them by equalities of the form ∥xi+1 − xi∥ 2 2 = (L… view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Circle packing. 5.0 2.5 0.0 2.5 5.0 6 4 2 0 2 4 6 0.60 0.65 0.70 0.75 coverages 0 10 20 30 40 50 60 frequency [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: The best circle packing found over 500 random initializations (left), and a histogram of the coverages obtained across all initializations (right). 26 [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: Car trajectory. Top. Position and orientation of the car. Middle. Speed and steering angle. Bottom. Acceleration and steering rate. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_6.png]
Figure 7
Figure 7. Figure 7: Location estimation from range measurements. Left. Range measurements without noise. Right. Range measurements with noise. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p033_8.png]
Figure 9
Figure 9. Figure 9: Phase retrieval. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_9.png]
Figure 10
Figure 10. Figure 10 [PITH_FULL_IMAGE:figures/full_fig_p038_10.png]
Figure 11
Figure 11. Figure 11: SVI model calibration. The red crosses show the observed implied volatilities, and the blue curve shows the fitted SVI model. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_11.png]
Figure 12
Figure 12. Figure 12: The percentage of held-out options for which the model price falls within the bid-ask spread, across all cross-sections. Results. We measure out-of-sample pricing accuracy using the root-mean-square relative pricing error (RMRPE), defined as RMRPE = vuut 1 N X N i=1 V…
Figure 13
Figure 13. Figure 13: Optimal power flow. 50 [PITH_FULL_IMAGE:figures/full_fig_p050_13.png]
Figure 14
Figure 14. Figure 14: Th´evenin battery model. The predicted terminal voltage at time tk is then vk = v oc k + R0 ik + U RC k , where v oc k = a + b/(Qcrit − qk). We wish to find the six parameters a, b, Qcrit , R0, R1, and C1 that best fit the measured terminal voltages in the least-squar…
Figure 15
Figure 15. Figure 15: Simulated battery data: current profile (top), stored charge (middle), and measured terminal voltage (bottom). 56 [PITH_FULL_IMAGE:figures/full_fig_p056_15.png]
Figure 16
Figure 16. Figure 16: Nonnegative matrix factorization for decomposing images into parts. 58 [PITH_FULL_IMAGE:figures/full_fig_p058_16.png]
Figure 17
Figure 17. Figure 17: Fitting an exponential decay model with outliers. # Huber variant cost = sum(huber(residuals, M)) prob = Problem(Minimize(cost)) prob.solve(nlp=True) Results. We generate m = 50 noisy measurements from the true model with pa￾rameters a = 5.0, λ = 0.30, and c = 1.0, ad…
Figure 18
Figure 18. Figure 18: Training data and decision boundary of neural network classifier for three values of the regularization parameter λ. Problem instance. We generate N = 200 perturbed samples from two interleav￾ing half-circles in R 2 , divided into 100 training and 100 test samples. Th…

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