Smooth Logic Constraints in Nonlinear Optimization and Optimal Control Problems

E. C. Kerrigan; J. Wehbeh

REVIEW 3 major objections 4 minor 25 references

Smooth Logic Constraints in Nonlinear Optimization and Optimal Control Problems

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A single reformulation turns arbitrary logic constraints into smooth math, removing the need for binary variables and mixed-integer solvers.

desk verdict The central exactness claim is false: Lemma 1's epsilon is ill-defined and the practical epsilon=0 changes the feasible set; the numerical examples also implement different logic than stated. read the letter →

arxiv 2506.01742 v2 pith:QXTG6D56 submitted 2025-06-02 eess.SY cs.SY

J. Wehbeh , E. C. Kerrigan This is my paper

classification eess.SYcs.SY MSC 90C3049M37

keywords logicconstraintsoptimalcontrolsmoothreformulationmax-minnonlinearprogrammingbinaryvariablesmixed-integerquadrotor

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 claims that any logic constraint built from equality and inequality propositions combined with NOT, AND, and OR can be rewritten as ordinary smooth constraints, so the resulting optimization problem has the same solution set and can be solved by a standard continuous solver. This matters because optimal control problems with if-then rules usually require binary variables and mixed-integer solvers, which are slower and scale poorly once the dynamics are nonlinear. The authors prove the reduction in three steps: simplify the logic to inequalities with AND and OR only, translate AND and OR into max and min operators, and replace the resulting max-min constraint by a differentiable constraint with auxiliary simplex variables. They demonstrate the method on two planar quadrotor tasks, reporting faster computation and more frequent convergence to the optimum than complementarity and big-M baselines.

What carries the argument

The machinery is the max-min reading of logic: an OR of propositions $p_j \le 0$ is equivalent to $\min_j p_j \le 0$, and an AND of clauses is equivalent to $\max_i \left(\min_j p_{i,j}\right) \le 0$. The paper then applies a known smoothing result to replace each inner min over a finite set by a weighted sum $\sum_j \lambda_j p_j$ with $\lambda$ on the standard simplex, making the whole logic constraint differentiable while preserving the solution set under the stated regularity assumption. Equality propositions are first converted into paired inequalities, and negations are pushed to the proposition level and removed by flipping the sign of the function, so the logic program reduces to conjunctive normal form in inequality propositions only.

What would settle it

Solve a one-variable problem whose only logic constraint is NOT(p <= 0) with p(x) = x, using the paper's replacement p_bar = -p. At x = 0 the original constraint is false, because p=0 satisfies p<=0 and its negation is false, while the smoothed constraint -0 <= 0 is true, so the smoothed problem accepts x=0 as feasible. If a run of the method returns such a boundary point that the original logic rejects, the claimed exactness of the solution set fails in that instance.

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Extended reading notes

Core claim

At the center is Theorem 1: for any logic program $L$ satisfying the paper's assumptions, the logical constraint $L(x,u)=true$ can be replaced by the family of smooth constraints $\sum_j \lambda_{i,j} p_{i,j}(x,u) \le 0$ for each conjunctive clause $i$, with each $\lambda$ vector on the simplex, without affecting the solution set. In plain terms, the paper claims an exact bridge from discrete logic to continuous differentiability: no binary variables, no relaxation penalty, and no loss of feasible solutions relative to the original logic-constrained problem. The price is the regularity assumption that the functions defining the propositions are twice differentiable and satisfy a non-degeneracy condition when gradients are tied together by a min operation.

Load-bearing premise

The load-bearing premise is that a strict inequality can be converted into a non-strict one by subtracting an arbitrarily small positive epsilon, but since no real epsilon is smaller than every positive delta, the negation step is exact only in the limit and can be wrong exactly on constraint boundaries.

Editorial extensions

If this is right

Any optimal control problem whose logic can be written with equality and inequality propositions plus NOT, AND, and OR can be handed to a standard continuous solver without binary variables.
For the two quadrotor benchmarks, the smoothed formulation finds the optimum in 81.3% of runs versus 66.0% for the best baseline (big-M), with lower average computation time and fewer infeasible runs.
Because the reformulation preserves the solution set exactly under the stated assumptions, feasible trajectories found by the smoothed problem satisfy the original logical specification, not merely an approximation.
The conversion to conjunctive normal form is not mandatory; direct min and max conversion may be numerically better in some problems, so the theorem provides a family of equivalent smooth formulations rather than a single fixed encoding.

Reading between the lines

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

Boundary cases need care beyond the paper's proof: at points where a negated proposition function vanishes, flipping the sign can make both the proposition and its negation appear true, so a practical implementation should treat boundary satisfaction as approximate or add a safety margin.
The smoothing variables are simplex weights, so their initialization may influence which local solution a nonconvex solver reaches; varying these initializations could be a practical tuning knob that the paper does not explore.
The same reduction applies to any mathematical program with logic constraints, not only dynamic optimal control, and could be tested on general mixed-integer nonlinear benchmark problems to compare against binary encodings.
The choice of representation, such as conjunctive normal form versus direct max-min conversion, changes the number of auxiliary variables and the curvature of the constraints; finding an optimal representation is left open and could be approached heuristically or empirically.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. The paper's headline claim—that any logic constraint can be replaced by a smooth constraint 'without affecting the solution set'—is false as stated. The counterexample in the stress test is correct: minimize x subject to ¬(x≤0) on [-1,1]. The original feasible set is (0,1], so no minimum is attained. The paper's replacement with ε=0 gives x≥0, and the NLP returns x=0, which violates the original logic. Lemma 1 requires 0<ε<δ for all δ>0, which no real ε satisfies, and the practical choice ε=0 turns ¬(p≤0) into p≥0, so the two overlap on the boundary. This is load-bearing: every negated inequality passes through Lemma 1, so Theorem 1 and the 'identical solution set' claim fail.

The second issue is that the numerical examples appear to implement different logic than stated. In Problem 1, (20) reads roughly as 'if not (green at both times 2 and 3) then not (red at all times 5-9),' but the smoothed constraint (21) is a set of constraints of the form λ1 A_2 + λ2 A_3 - λ3 B_i ≤0 for each i, which corresponds to 'green at time 2 or 3, or avoid red at every i'. Those are not the same condition. So the performance numbers do not validate the method as described.

What's worth keeping: the paper does a clean job assembling the standard logic-to-max-min equivalences (Lemma 2 is correct), and drawing attention to Kirjner-Neto and Polak's smoothing theorem is useful. The empirical comparison with complementarity and Big-M on two quadrotor problems is a reasonable effort, and the speedups are plausible if the constraints were encoded correctly. The authors also honestly note the method was used in their prior work.

The soft spots are not minor, but they are fixable in principle. The boundary issue is not a measure-zero artifact; an optimizer can select the boundary. The proof of Theorem 1 is deferred to [19], which is fine for the smoothing step, but it does not address the negation step that Lemma 1 botches. To make the paper sound, the authors would need to either restrict to problems with interior solutions and use a fixed ε>0, or explicitly present the method as an approximation, and then correct the logic statements in the examples.

Who this is for: researchers in continuous encoding of logic for optimal control. They may get useful ideas from the max-min smoothing, but should not rely on the exactness claim. The paper deserves referee time because the underlying idea is worth repairing, but in its current form it should be rejected. I would send it to peer review with the expectation of heavy revision and a re-review after the negation issue is addressed.

Referee Report

3 major / 4 minor

Summary. The paper proposes a method for reformulating logic constraints in optimal control problems as smooth continuous constraints, avoiding binary variables or mixed-integer solvers. The approach first rewrites any logic program built from equality and inequality propositions with NOT, AND, and OR into conjunctive normal form, then converts the resulting AND/OR structure into max-min constraints, and finally smooths the max-min constraints using auxiliary simplex variables, citing a result of Kirjner-Neto and Polak. The central claim, stated in Theorem 1 and Section III-D, is that the reformulation has an identical solution set to the original logic-constrained problem. The method is demonstrated on two planar quadrotor optimal control problems with conditional logic constraints, with numerical comparisons against complementarity and Big-M formulations.

Significance. If the central exactness claim were correct, the paper would offer a practically valuable bridge between discrete logic and continuous, derivative-based optimization, and the numerical comparisons on the quadrotor examples are informative. The paper is clearly written and the numerical setup is reproducible in principle, with randomized restarts and timing statistics. However, the load-bearing theoretical claim, namely that the smooth reformulation preserves the solution set exactly, rests on the negation-elimination step of Lemma 1, and that step is not mathematically sound. Because the exactness claim is the stated contribution, the flaw affects the main result of the paper.

major comments (3)

[III-A, Lemma 1] The elimination of negated inequalities via Eq. (10b) requires an epsilon with 0 < epsilon < delta for every delta > 0; no real number has this property. The paper then states that in practice setting ̄p_i = -p_i is sufficient, but with epsilon = 0 the replacement turns ¬(p ≤ 0) into p ≥ 0. At p = 0, both p ≤ 0 and its replacement hold, so the claimed logical equivalence fails on constraint boundaries. This is not a harmless measure-zero issue: for the problem minimize x subject to ¬(x ≤ 0) and x ∈ [-1,1], the original feasible set is (0,1] and no minimum is attained, while the smoothed replacement p ≥ 0 admits x = 0, a point that violates the original logic. Since every negated inequality in the CNF of Section III-B passes through this step, the error propagates into Theorem 1 and the 'identical solution set' claim of Section III-D.
[III-C, Theorem 1] The proof of Theorem 1 is deferred to [19] and does not address the negation-elimination step that precedes Lemma 2. Even accepting the max-min-to-simplex conversion from [19], that conversion can only preserve the solution set of the max-min constraint (12); it cannot repair the fact that the max-min constraint itself is derived using Lemma 1's inexact treatment of negated inequalities. The manuscript also asserts Assumption 3.9 of [19] without checking it for the two quadrotor examples, so the numerical demonstrations do not validate the hypotheses under which the external theorem is invoked.
[III-A, paragraph after Eq. (10)] The statement that solvers cannot distinguish strict and non-strict inequalities explains why an epsilon-perturbation might be useful numerically, but it does not justify the exactness claim. The paper's contribution is explicitly stated as replacing the logic constraint 'without affecting the solution set' and producing an 'identical solution set' in Section III-D. As the counterexample above shows, boundary points where a proposition and its negation overlap can be selected by a continuous optimizer, so the practical recipe with epsilon = 0 changes the feasible set. A correct treatment would need to handle strict inequalities explicitly, which standard NLP solvers cannot do, or would require weakening the exactness claim substantially.

minor comments (4)

[IV-C, first paragraph] The text says the optimal solution to Problem 1 is 'as seen in Figure 3 (c)', but the corresponding trajectory is shown in Figure 2 (c); this cross-reference should be corrected.
[Eq. (21)] The displayed smoothed constraint for Problem 1 is ambiguous: it is not clear whether a separate constraint is generated for each i ∈ {5,...,9} or whether the expression is a single constraint, and the λ variables are not indexed by i even though they should belong to different simplex sets for different i.
[II, Eq. (3)] The domain notation f : R^{n/(N+1)} × R^{m/N} → R^{n/(N+1)} is dimensionally opaque; the standard notation f : R^n × R^m → R^n with time indices would be clearer.
[Abstract and Introduction] The paper repeatedly claims to handle 'arbitrary logic constraints', but Assumption 2 restricts to twice-differentiable constituent functions with a non-degeneracy condition. This restriction should be stated in the abstract and introduction so that the scope of the claim is accurate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reformulation chain rests on external standard results, not on self-citation or fitted inputs.

full rationale

The derivation chain is: Lemma 1 reduces equality propositions to inequality conjunctions using explicit algebraic constructions (4)-(5), then eliminates negations via De Morgan equivalences (6)-(8) and the replacement (10). Lemma 2 converts AND/OR to max/min using the standard logical identities quoted from [22] and CNF from [21]. Theorem 1 then invokes an external smoothing theorem, Kirjner-Neto and Polak [19], to replace each finite max-min constraint by smooth constraints (13); the proof of Theorem 1 is explicitly 'apply the smoothing approach of [19]', which is not authored by the present paper and is not itself derived from the paper's claims. Assumption 2 likewise defers to Assumption 3.9 in [19]. No parameter is fitted to data and no quantity that is predicted is an input by construction. The self-citations [20] and [23] are provenance and illustration only: [20] states the method 'was previously utilized in our work', and [23] is only the source of the quadrotor figure. Consequently there is no exhibited reduction of a claimed theorem to its own assumptions. The one genuinely questionable step is Lemma 1's epsilon condition (0<epsilon<delta for every delta>0 in R), which no real epsilon satisfies, and the practical choice epsilon=0 makes the replacement of negated inequalities approximate on the boundary p=0. That is a correctness/technical flaw in the claimed exactness, not a circular definition: it is not defined in terms of the target solution set, fitted to outputs, or justified exclusively by a self-citation. Under the hard rule that circularity must be exhibited as Eq. X = Eq. Y by construction or a self-citation chain, that flaw is outside the circularity count. The central result therefore has independent content and is benchmarked against external methods [24,25]; 0 is the appropriate score.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The only ad hoc quantity is epsilon, which is not a real number and is replaced by 0 in practice. The auxiliary simplex variables lambda are optimization variables, not fitted parameters. The method postulates no new physical entities.

free parameters (1)

epsilon = 0 (practice); no real value satisfies 0<epsilon<delta for all delta>0
Introduced in Lemma 1 to convert strict inequalities p>0 into p-bar <= 0; undefined in standard reals and set to 0 in practice, which alters the logic at p=0.

assumptions (6)

standard math Functional completeness of NOT, AND, OR for finite propositional logic
Used to justify that L can express any logic (Section II, Assumption 1).
standard math Every propositional formula can be put in conjunctive normal form without changing truth
Used in Lemma 2 to transform L into (11).
standard math For real numbers, AND of inequalities is max and OR of inequalities is min
Cited from [22], used for Eq. (12).
domain assumption Assumption 1: logic consists of finitely many equality/inequality propositions linked by NOT, AND, OR
Defines the class of problems considered; excludes temporal operators and quantifiers over infinite sets.
domain assumption Assumption 2: functions twice differentiable and gradients satisfy Assumption 3.9 of [19]
Required for Theorem 1 to apply [19]; unverified in the examples and restrictive.
ad hoc to paper Solvers cannot distinguish strict and non-strict inequalities, so epsilon can be dropped
Used to justify replacing NOT(p<=0) with -p<=0; changes boundary behavior.

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Cite this review

Pith. "Pith review of Smooth Logic Constraints in Nonlinear Optimization and Optimal Control Problems." pith.science (2026). https://pith.science/paper/QXTG6D56

@misc{pith2026250601742,
  author       = {Pith},
  title        = {Pith review of: Smooth Logic Constraints in Nonlinear Optimization and Optimal Control Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXTG6D56}},
  note         = {Machine review of arXiv:2506.01742}
}

read the original abstract

In some optimal control problems, complex relationships between states and inputs cannot be easily represented using continuous constraints, necessitating the use of discrete logic instead. This paper presents a method for incorporating such logic constraints directly within continuous optimization frameworks, eliminating the need for binary variables or specialized solvers. Our approach reformulates arbitrary logic constraints under minimal assumptions as max-min constraints, which are then smoothed by introducing auxiliary variables into the optimization problem. When these reformulated constraints are satisfied, they guarantee that the original logical conditions hold, ensuring correctness in the optimization process. We demonstrate the effectiveness of this method on two planar quadrotor control tasks with complex logic constraints. Compared to existing techniques for encoding logic in continuous optimization, our approach achieves faster computational performance and improved convergence to feasible solutions.

Figures

Figures reproduced from arXiv: 2506.01742 by the authors.

**Figure 2.** Obstacle (red), logic trigger (green), target (black), and solution trajectory (blue) for Problem 1. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Obstacle (red), logic trigger (green), targets (black x and green +), and solution trajectory (blue) for Problem 2. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [19]

On the conversion of optimization problems with max-min constraints to standard optimization prob- lems,

C. Kirjner-Neto and E. Polak, “On the conversion of optimization problems with max-min constraints to standard optimization prob- lems,” SIAM Journal on Optimization , vol. 8, no. 4, pp. 887–915, 1998

work page 1998

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R. B. Vinter, Optimal control. Springer, 2010, vol. 2, no. 1

work page 2010

[3] [2]

Nocedal and S

J. Nocedal and S. J. Wright, Numerical optimization. Springer, 2006. (a) Going through the green circle allows passing through the red obstacle. (b) Sample sub-optimal trajectory. (c) Optimal solution trajectory. Fig. 2. Obstacle (red), logic trigger (green), target (black), and solution trajectory (blue) for Problem 1. (a) Going through the green circle ...

work page 2006

[4] [3]

Optimization-based control for dynamic legged robots,

P. M. Wensing, M. Posa, Y . Hu, A. Escande, N. Mansard, and A. Del Prete, “Optimization-based control for dynamic legged robots,” IEEE Transactions on Robotics , vol. 40, pp. 43–63, 2023

2023

[5] [4]

Robust optimal control for demand side management of multi- carrier microgrids,

R. Carli, G. Cavone, T. Pippia, B. De Schutter, and M. Dotoli, “Robust optimal control for demand side management of multi- carrier microgrids,” IEEE Transactions on Automation Science and Engineering, vol. 19, no. 3, pp. 1338–1351, 2022

work page 2022

[6] [5]

Fast homotopy for spacecraft ren- dezvous trajectory optimization with discrete logic,

D. Malyuta and B. Ac ¸ıkmes ¸e, “Fast homotopy for spacecraft ren- dezvous trajectory optimization with discrete logic,” Journal of Guid- ance, Control, and Dynamics , vol. 46, no. 7, pp. 1262–1279, 2023

work page 2023

[7] [6]

A review and comparison of solvers for convex MINLP,

J. Kronqvist, D. E. Bernal, A. Lundell, and I. E. Grossmann, “A review and comparison of solvers for convex MINLP,” Optimization and Engineering, vol. 20, pp. 397–455, 2019

work page 2019

[8] [7]

Undecidability and hardness in mixed-integer nonlinear programming,

L. Liberti, “Undecidability and hardness in mixed-integer nonlinear programming,” RAIRO-Operations Research, vol. 53, no. 1, pp. 81– 109, 2019

work page 2019