Sign conventions across the ecosystem: min/max, p⁰, inequality constraints, signs of p and of the duals

[ocots]

Sign conventions across the ecosystem: min vs max, p⁰, inequality constraints, and the signs of p and of the duals

This is a request to think about and settle one convention for the whole ecosystem (CTDirect, CTFlows, CTModels, OptimalControl docs). Nothing is decided here; the aim is to list the choices, their consequences, and what the code does today.

Why now

Three pieces of work all depend on these signs:

• Flow(ocp, law) Solution: reconstruct the constraint duals from the multipliers CTFlows.jl#472: Flow(ocp, law) should return the constraint duals, comparable with the ones of a direct solve. Its sign rule depends on the choices below.
• Box multipliers are not divided by the time step, and their sign is the opposite of the path multipliers CTDirect.jl#639: box duals are not divided by the step and have the opposite sign to the path duals; it proposes to choose one convention.
• control-toolbox/CTFlows.jl discussion 303 present ct repl in the documentation #312 (design note, July 2026): multipliers are inputs of the constrained flow (H̃ + μ·g); the label is the key that links them to dual(sol, ocp, label).

Past related threads: control-toolbox/CTDirect.jl#515 and #663 (for max f(x) Ipopt returned f and MadNLP -f; solved on the CTDirect side), control-toolbox/CTDirect.jl#405 (scaling of the path duals).

What the code does today (probed, 2026-10-21)

Probes with OptimalControl 2.2.1-beta (CTDirect 1.1.8-beta) with Ipopt and MadNLP, and CTFlows main, on problems with a closed form. A constraint is written g = sign·(c − bound) ≥ 0 (sign = +1 lower side or equality, −1 upper side), and the CTFlows pseudo-Hamiltonian is H̃ = p·f + sp0·ℓ + μ·g with sp0 = −1 for :min, +1 for :max.

	CTFlows	CTDirect
:min, path constraint	μ ≥ 0 for g ≥ 0	dual = −sign·μ (for all sides, Ipopt = MadNLP)
:min, box constraint	same μ	dual = −sign·μ·h (not divided by h; opposite sign to a path constraint)
:max instead of :min (same problem written −ℓ)	same p and μ: H̃ is identical (sp0·ℓ is unchanged), and the control is still argmax H̃	p and the duals change sign (p⁰ stays −1, so H = p·f − ℓ)

On the problem ẋ = u, x(0) = −1, x + u ≤ 0, ∫(−u) → min (and the same with ∫u → max): CTDirect gives p = −0.63… in :min and +0.63… in :max, and dual(:mixed) = ±e^{t−1} respectively. CTFlows gives the same p for both.

Questions to settle

1. min and max. If the user changes min J into max −J, should p and the duals change sign or not?
   • Yes (sensitivity convention, as in NLP: the dual is the derivative of the optimal value of the criterion as written; p⁰ = −1 always, H = p·f − ℓ, and max means argmin H). It is the convention of CTDirect today.
   • No (geometric convention, H is always maximised; sp0 = s·p⁰ flips with the criterion). It is the convention of CTFlows today, and the control law u = argmax H does not depend on the criterion.
2. Value of p⁰. −1 (OptimalControl docs, "Notation and conventions") with a sign s for min/max, or +1? Is p⁰ normalised, or free (abnormal extremals)?
3. Hamiltonian and inequality constraints. H̃ = p·f + p⁰ℓ + μ·c with c ≥ 0 and μ ≥ 0, or c ≤ 0 and μ ≥ 0 (written μ·g with g ≤ 0 in the older CTFlows)? And for a two-sided constraint lb ≤ c ≤ ub: two multipliers μ_l, μ_u ≥ 0 with μ = μ_l − μ_u (as in CTDirect#639), or the active side given explicitly (as constraint = :label => :upper in CTFlows)?
4. Equalities. One signed multiplier; which sign convention, and is it the one of the inequalities on the lower side?
5. Transversality and boundary constraints. p(t_f) = ∓∇g + ∂b'β: sign of β and of the dual of a boundary constraint (CTDirect#639: dual(:target) = −p(t_f) today).
6. Same sign for the types of constraint? A path, a box and a boundary constraint should give the same sign for the same constraint written in the three forms (today path and box are opposite).
7. Scaling. Duals as densities (divided by the quadrature weight, a continuous-time multiplier) or as raw NLP values? Needed to compare direct and indirect methods.
8. Solvers. The convention must not depend on the NLP solver (the old Ipopt/MadNLP max gap, now gone): which tests keep this true?

What is asked

Please give your view on 1, 2 and 3 first (the others follow). I propose to decide, then:

• state the convention once, in one page of the OptimalControl documentation (see the linked docs issue);
• align CTDirect (Box multipliers are not divided by the time step, and their sign is the opposite of the path multipliers CTDirect.jl#639) and CTFlows (Flow(ocp, law) Solution: reconstruct the constraint duals from the multipliers CTFlows.jl#472) on it, and add a test that compares dual(sol, ocp, label) of a direct and of an indirect solve;
• decide whether p flips with min/max and, if so, change it in CTFlows, which affects the sign of p0 in shooting functions written for :max problems.

[ocots]

Documentation page, once the choices are made: #1015. Tracking issues: control-toolbox/CTFlows.jl#472, control-toolbox/CTDirect.jl#639.

[PierreMartinon]

As a remark, on the direct side, the min/max question is deferred to the NLP modeler, so we get multipliers according to their implementation. This of course still allows us to perform post treatment.

Do we know what convention other optimisation tools use ?

[ocots]

Actually, the sign convention must be set in CTSolvers, not in CTDirect. But the scaling of the duals should be in CTDirect, cf. control-toolbox/CTDirect.jl#639.

[ocots]

Where the sign of the duals will be implemented: control-toolbox/CTSolvers.jl#288 (solver-independent convention for the multipliers; evidence in control-toolbox/CTSolvers.jl#287). The scaling by the node weights stays in control-toolbox/CTDirect.jl#639.

[ocots]

Multiplier signs of the NLP solvers: min f vs max -f, bound vs constraint row

Generated by probe/signs/probe_signs.jl of CTSolvers on 2026-10-23 (Julia 1.12.6).

Package	Version
CTSolvers	0.6.2
NLPModels	0.21.12
ADNLPModels	0.8.15
ExaModels	0.12.1
NLPModelsIpopt	0.11.3
MadNLP	0.10.2
MadNCL	0.2.3
UnoSolver	0.3.9
Ipopt_jll	300.1400.1902+0
Uno_jll	2.9.2+0

Problem

F = f  (min)   or   F = -f  (max),   f(x) = (x₁ - 3)² + (x₂ - 1)² - x₃ + x₄ + 2x₅
c(x) = x₁ + x₂ - 3 = 0
g(x) = x₁² + x₂² - 6 ≤ 0           written  g ≤ 0   or   -g ≥ 0      (active)
x₃ ≤ 1                             written  as a bound, or as the row -x₃ ≥ -1   (active)
x₄ ≥ 0                             a bound   (active)
x₅ = 1/2                           a fixed variable (lvar = uvar)
x* = ((3 + √3)/2, (3 - √3)/2, 1, 0, 1/2)

Eight ways of writing it (max -f is the same problem as min f):

#	Case
1	min f · g ≤ 0 · x₃ ≤ a
2	min f · -g ≥ 0 · x₃ ≤ a
3	max -f · g ≤ 0 · x₃ ≤ a
4	max -f · -g ≥ 0 · x₃ ≤ a
5	min f · g ≤ 0 · -x₃ ≥ -a
6	min f · -g ≥ 0 · -x₃ ≥ -a
7	max -f · g ≤ 0 · -x₃ ≥ -a
8	max -f · -g ≥ 0 · -x₃ ≥ -a

Notation and conventions

F is the objective as written in the model (F = -f for max -f), and the solvers maximise it by minimising -F. y gathers the multipliers of the rows lcon ≤ c(x) ≤ ucon, i.e. of the equality c, of the inequality g and, in the row form, of -x₃ ≥ -a: one signed value per row, y = μU − μL, where μL, μU are the multipliers of the lower and upper side of the row, at most one being nonzero. So Jᵀy = Jᵀ(μU − μL), J being the Jacobian of the rows, and an equality has a single signed y. zL and zU are the same thing for the bounds of the variables lvar ≤ x ≤ uvar, stored in two vectors; x₅, fixed, has a single net multiplier.

A convention is the choice of three things:

1. the stationarity of the Lagrangian at the solution, for F as written, i.e. the signs of the multipliers in ∇F + Jᵀy ± zL + zU = 0;
2. the dual feasibility, i.e. which multipliers are nonnegative for a minimisation (μL, μU, zU, and zL or -zL according to its sign in the stationarity);
3. the rule for a maximisation: the solver reports either the multipliers of F as written (as written: every multiplier changes sign with the sense, and so does dual feasibility), or those of the equivalent minimisation min(-F) (min-equiv.: nothing changes sign). For a minimisation the two rules coincide.

That gives four conventions:

Convention	Stationarity and dual feasibility (minimisation)	Maximisation
A, as written	∇F + Jᵀy − zL + zU = 0, with μL, μU, zL, zU ≥ 0 for a minimisation	multipliers of F as written: all flip
A, min-equiv.	∇F + Jᵀy − zL + zU = 0, with μL, μU, zL, zU ≥ 0 for a minimisation	multipliers of min(-F): none flips
B, as written	∇F + Jᵀy + zL + zU = 0, with μL, μU, zU ≥ 0 and zL ≤ 0 for a minimisation	multipliers of F as written: all flip
B, min-equiv.	∇F + Jᵀy + zL + zU = 0, with μL, μU, zU ≥ 0 and zL ≤ 0 for a minimisation	multipliers of min(-F): none flips

With the active set above the multipliers are unique, so each convention gives one expected value per multiplier, computed from the gradient and Jacobian of the model at x*. A solver follows a convention if all its multipliers match.

Expected values

Convention A, as written

#	y_c (equality)	y_g (inequality)	x₃	zL(x₄)	net(x₅)
1	+0.5359	+0.1547	zU(x₃) = +1.0000	+1.0000	-2.0000
2	+0.5359	-0.1547	zU(x₃) = +1.0000	+1.0000	-2.0000
3	-0.5359	-0.1547	zU(x₃) = -1.0000	-1.0000	+2.0000
4	-0.5359	+0.1547	zU(x₃) = -1.0000	-1.0000	+2.0000
5	+0.5359	+0.1547	y₃ = -1.0000	+1.0000	-2.0000
6	+0.5359	-0.1547	y₃ = -1.0000	+1.0000	-2.0000
7	-0.5359	-0.1547	y₃ = +1.0000	-1.0000	+2.0000
8	-0.5359	+0.1547	y₃ = +1.0000	-1.0000	+2.0000

Convention A, min-equiv.

#	y_c (equality)	y_g (inequality)	x₃	zL(x₄)	net(x₅)
1	+0.5359	+0.1547	zU(x₃) = +1.0000	+1.0000	-2.0000
2	+0.5359	-0.1547	zU(x₃) = +1.0000	+1.0000	-2.0000
3	+0.5359	+0.1547	zU(x₃) = +1.0000	+1.0000	-2.0000
4	+0.5359	-0.1547	zU(x₃) = +1.0000	+1.0000	-2.0000
5	+0.5359	+0.1547	y₃ = -1.0000	+1.0000	-2.0000
6	+0.5359	-0.1547	y₃ = -1.0000	+1.0000	-2.0000
7	+0.5359	+0.1547	y₃ = -1.0000	+1.0000	-2.0000
8	+0.5359	-0.1547	y₃ = -1.0000	+1.0000	-2.0000

Convention B, as written

#	y_c (equality)	y_g (inequality)	x₃	zL(x₄)	net(x₅)
1	+0.5359	+0.1547	zU(x₃) = +1.0000	-1.0000	-2.0000
2	+0.5359	-0.1547	zU(x₃) = +1.0000	-1.0000	-2.0000
3	-0.5359	-0.1547	zU(x₃) = -1.0000	+1.0000	+2.0000
4	-0.5359	+0.1547	zU(x₃) = -1.0000	+1.0000	+2.0000
5	+0.5359	+0.1547	y₃ = -1.0000	-1.0000	-2.0000
6	+0.5359	-0.1547	y₃ = -1.0000	-1.0000	-2.0000
7	-0.5359	-0.1547	y₃ = +1.0000	+1.0000	+2.0000
8	-0.5359	+0.1547	y₃ = +1.0000	+1.0000	+2.0000

Convention B, min-equiv.

#	y_c (equality)	y_g (inequality)	x₃	zL(x₄)	net(x₅)
1	+0.5359	+0.1547	zU(x₃) = +1.0000	-1.0000	-2.0000
2	+0.5359	-0.1547	zU(x₃) = +1.0000	-1.0000	-2.0000
3	+0.5359	+0.1547	zU(x₃) = +1.0000	-1.0000	-2.0000
4	+0.5359	-0.1547	zU(x₃) = +1.0000	-1.0000	-2.0000
5	+0.5359	+0.1547	y₃ = -1.0000	-1.0000	-2.0000
6	+0.5359	-0.1547	y₃ = -1.0000	-1.0000	-2.0000
7	+0.5359	+0.1547	y₃ = -1.0000	-1.0000	-2.0000
8	+0.5359	-0.1547	y₃ = -1.0000	-1.0000	-2.0000

The expected values satisfy the dual feasibility of their convention in every case: true.

What each solver returns

A cell is the convention the solver follows when all its multipliers match it: A or B for a minimisation (the two rules for a maximisation coincide there), and A, as written / A, min-equiv. (or B) for a maximisation. If they match none of the considered conventions, the cell is none, and the multipliers that disagree are listed below. ⚠ status marks a solve that did not converge.

ADNLPModels

Solver	1	2	3	4	5	6	7	8
Ipopt	A	A	A, as written	A, as written	A	A	A, as written	A, as written
MadNLP	A	A	none	none	A	A	none	none
Uno	B	B	B, as written	B, as written	B	B	B, as written	B, as written
CTSolvers.Ipopt	A	A	A, as written	A, as written	A	A	A, as written	A, as written
CTSolvers.MadNLP	A	A	none	none	A	A	none	none
CTSolvers.MadNCL	A	A	none	none	A	A	none	none
CTSolvers.Uno	B	B	B, as written	B, as written	B	B	B, as written	B, as written

ExaModels

Solver	1	2	3	4	5	6	7	8
Ipopt	A	A	A, as written	A, as written	A	A	A, as written	A, as written
MadNLP	A	A	none	none	A	A	none	none
Uno	B	B	B, as written	B, as written	B	B	B, as written	B, as written
CTSolvers.Ipopt	A	A	A, as written	A, as written	A	A	A, as written	A, as written
CTSolvers.MadNLP	A	A	none	none	A	A	none	none
CTSolvers.MadNCL	A	A	none	none	A	A	none	none
CTSolvers.Uno	B	B	B, as written	B, as written	B	B	B, as written	B, as written

Does the multiplier change sign from min f to max -f?

Independent of any convention: the same problem is written as min f and as max -f (cases 1/3, 2/4, 5/7, 6/8), and each multiplier of the second is compared with the one of the first. A solver that reports the multipliers of the objective as written should show flips everywhere (the objective is -f, so every multiplier changes sign). same means the multiplier is reported for the equivalent minimisation. Read it together with the verdicts above: a solver whose bound multipliers do not flip can look like the other convention on some cases.

Solver	rows y	upper bound zU(x₃)	lower bound zL(x₄)	fixed variable, net(x₅)
Ipopt	flips	flips	flips	flips
MadNLP	flips	same	same	same
Uno	flips	flips	flips	flips
CTSolvers.Ipopt	flips	flips	flips	flips
CTSolvers.MadNLP	flips	same	same	same
CTSolvers.MadNCL	flips	same	same	same
CTSolvers.Uno	flips	flips	flips	flips

Same on both modelers, and CTSolvers vs the raw solver

• ADNLPModels and ExaModels give the same verdicts: true.
• CTSolvers.Ipopt returns the same multipliers as the raw Ipopt: true.
• CTSolvers.MadNLP returns the same multipliers as the raw MadNLP: true.
• CTSolvers.Uno returns the same multipliers as the raw Uno: true.

Multipliers that disagree with the conventions

For the cases where a solver follows none of the conventions (ADNLPModels; ExaModels is identical), its multipliers next to the expected ones for its own stationarity convention (the one it follows on every minimisation), under both rules for a maximisation. The solver's value is in bold when it matches neither rule.

MadNLP (stationarity A)

#	multiplier	solver	A, as written	A, min-equiv.
3	y_c	-0.5359	-0.5359	+0.5359
3	y_g	-0.1547	-0.1547	+0.1547
3	z_x3	+1.0000	-1.0000	+1.0000
3	z_x4	+1.0000	-1.0000	+1.0000
3	z_x5	-2.0000	+2.0000	-2.0000
4	y_c	-0.5359	-0.5359	+0.5359
4	y_g	+0.1547	+0.1547	-0.1547
4	z_x3	+1.0000	-1.0000	+1.0000
4	z_x4	+1.0000	-1.0000	+1.0000
4	z_x5	-2.0000	+2.0000	-2.0000
7	y_c	-0.5359	-0.5359	+0.5359
7	y_g	-0.1547	-0.1547	+0.1547
7	y_x3	+1.0000	+1.0000	-1.0000
7	z_x4	+1.0000	-1.0000	+1.0000
7	z_x5	-2.0000	+2.0000	-2.0000
8	y_c	-0.5359	-0.5359	+0.5359
8	y_g	+0.1547	+0.1547	-0.1547
8	y_x3	+1.0000	+1.0000	-1.0000
8	z_x4	+1.0000	-1.0000	+1.0000
8	z_x5	-2.0000	+2.0000	-2.0000

CTSolvers.MadNCL (stationarity A)

#	multiplier	solver	A, as written	A, min-equiv.
3	y_c	-0.5359	-0.5359	+0.5359
3	y_g	-0.1547	-0.1547	+0.1547
3	z_x3	+1.0000	-1.0000	+1.0000
3	z_x4	+1.0000	-1.0000	+1.0000
3	z_x5	-2.0000	+2.0000	-2.0000
4	y_c	-0.5359	-0.5359	+0.5359
4	y_g	+0.1547	+0.1547	-0.1547
4	z_x3	+1.0000	-1.0000	+1.0000
4	z_x4	+1.0000	-1.0000	+1.0000
4	z_x5	-2.0000	+2.0000	-2.0000
7	y_c	-0.5359	-0.5359	+0.5359
7	y_g	-0.1547	-0.1547	+0.1547
7	y_x3	+1.0000	+1.0000	-1.0000
7	z_x4	+1.0000	-1.0000	+1.0000
7	z_x5	-2.0000	+2.0000	-2.0000
8	y_c	-0.5359	-0.5359	+0.5359
8	y_g	+0.1547	+0.1547	-0.1547
8	y_x3	+1.0000	+1.0000	-1.0000
8	z_x4	+1.0000	-1.0000	+1.0000
8	z_x5	-2.0000	+2.0000	-2.0000

Conclusion

Solver	Stationarity (read on min f)	Maximisation (max -f)
Ipopt	A	A, as written — every multiplier changes sign
MadNLP	A	none, inconsistent: the rows follow as written, the bounds and fixed variable min-equiv.
Uno	B	B, as written — every multiplier changes sign
CTSolvers.Ipopt	A	A, as written — every multiplier changes sign
CTSolvers.MadNLP	A	none, inconsistent: the rows follow as written, the bounds and fixed variable min-equiv.
CTSolvers.MadNCL	A	none, inconsistent: the rows follow as written, the bounds and fixed variable min-equiv.
CTSolvers.Uno	B	B, as written — every multiplier changes sign

• Stationarity A: Ipopt, MadNLP, CTSolvers.Ipopt, CTSolvers.MadNLP, CTSolvers.MadNCL.
• Stationarity B: Uno, CTSolvers.Uno.

A solver marked none has the stationarity of its family on min f, but on max -f it mixes the two rules: some multipliers change sign between min f and max -f (those of the rows), others do not (those of the bounds and of fixed variables). No single convention, as defined above, describes its output.

[ocots]

Proposed convention

Principle: a multiplier belongs to a constraint and to a problem, not to the way they are written.

1. Inequality constraints (rows l ≤ c(x) ≤ u and bounds lvar ≤ x ≤ uvar): every multiplier is nonnegative, for a minimisation and for a maximisation, whatever the constraint. There is one multiplier per active side: μL ≥ 0 for c ≥ l, μU ≥ 0 for c ≤ u (and zL, zU ≥ 0 for the bounds of the variables). In particular g ≥ 0 and -g ≤ 0 are the same constraint and have the same multiplier, with the same sign.
2. Equality constraints: no positivity condition, and the sign is not changed: it is the sign of the stationarity ∇F + Jᵀy − zL + zU = 0 (the convention A of the probe), the one of the costate p of the dynamics. An equality has no preferred side, so c = 0 and -c = 0 differ by a sign, as they must.
3. min f and max -f are the same problem, so they have the same multipliers: no sign changes between the two (the multipliers of the equivalent minimisation, "min-equiv." in the probe).

Stationarity, for the equivalent minimisation of F_min (= f): ∇F_min + Jᵀ(μU − μL) − zL + zU = 0, with μL, μU, zL, zU ≥ 0, at most one of μL, μU nonzero on a row, and the equality rows carrying their signed y.

What it gives on the cases of the probe

The 8 ways of writing the problem of control-toolbox/CTSolvers.jl#287 (min f / max -f, g ≤ 0 / -g ≥ 0, x₃ ≤ a as a bound / -x₃ ≥ -a as a row) collapse to one set of numbers:

Equality c = 0	Inequality g	x₃ ≤ a, bound or row	x₄ ≥ 0	Fixed x₅ (net)
y = +0.5359	μ = +0.1547	μ = +1	z = +1	−2 (no sign condition)

With the signed y of a row, as the solvers return it, g ≤ 0 gives +0.1547 and -g ≥ 0 gives −0.1547; the bound and the row form of x₃ ≤ a give +1 and −1. So the proposal is not one of the four conventions of the probe, which all report a signed y per row: no solver follows it as is, Ipopt included, and the multipliers of the inequality rows have to be split by active side.

The same convention in the indirect method: always maximise the pseudo-Hamiltonian

The maximum principle, as written in the OptimalControl documentation (Formulation → Notation and conventions), reads

H(t, x, p, u, v) = p·f(t, x, u, v) + p⁰ f⁰(t, x, u, v) + μ·g(t, x, u, v),     g ≥ 0, μ ≥ 0,

and along an optimal trajectory the control maximises H, for min f as for max -f. The only thing that depends on the sense of the criterion is p⁰, the multiplier associated with the criterion: p⁰ ≤ 0 for a minimisation (normal case p⁰ = −1) and p⁰ ≥ 0 for a maximisation (normal case p⁰ = +1); p⁰ = 0 is the abnormal case for both. (In the code of CTFlows this is sp0 = s·p⁰ with p⁰ = −1 and s = +1 for :min, s = −1 for :max.)

Then p⁰ f⁰ is the same for min f (p⁰ = −1, f⁰ = f) and for max -f (p⁰ = +1, f⁰ = −f), so H, the control that maximises it, the extremals, the costate p and the multipliers μ ≥ 0 of the constraints are the same. Only the dual associated with the criterion, p⁰, changes sign. Checked with CTFlows on ẋ = −x + u, min ∫ u²/2 against max ∫ −u²/2, with the same feedback u = p: the same x(1) = 0.72044 and p(1) = 0.81548, and an objective of opposite sign (it is the criterion).

This is the indirect counterpart of the rule of the direct method: there, max -f is treated as the equivalent minimisation min f (same Lagrangian, same multipliers); here, H is maximised in both cases and p⁰ carries the sign. The costate is the multiplier of the dynamics, an equality, so it keeps its sign in both methods.

The other possible convention, the multipliers of the criterion as written, would keep p⁰ = −1 for a maximisation: H = p·f − f⁰, the control would then minimise H, p and the duals would change sign with the sense, and the maximum principle would depend on it. This is what CTDirect does today for a maximisation (measured on the first probe, ẋ = u, x(0) = −1: p(0) = −0.63 for min ∫(−u), +0.63 for max ∫u).

Consequences

• CTFlows: it already follows the section above (sp0 = ±1, the control maximises H for min and max). Flow(ocp, law; constraint=:label => side, multiplier=μ) already normalises the constraint as g ≥ 0 and takes μ ≥ 0, so dual(sol, ocp, :label) is exactly that μ (and 0 off the arc), with no sign to juggle: this simplifies Flow(ocp, law) Solution: reconstruct the constraint duals from the multipliers CTFlows.jl#472. The costate p does not change with min/max.
• CTDirect: for a maximisation it currently reports the multipliers and the costate as written (they flip, p⁰ stays −1); with this convention they would not flip. The box and path duals get the same sign (Box multipliers are not divided by the time step, and their sign is the opposite of the path multipliers CTDirect.jl#639).
• CTModels: path_constraints_dual holds one signed value per row. For a single-sided inequality the dual is then μ ≥ 0 (the side is known from the bounds); for a two-sided row, to settle (signed value, or a pair).

[ocots]

The convention is defined in ct-conventions

The convention discussed here is now written as rules SGN1–SGN6 of ct-conventions: rules/signs.md. It is the reference to cite. Until control-toolbox/ct-conventions#49 is merged, the file is on its branch: rules/signs.md (feat/sign-conventions).

Rule	Content
SGN1	min f and max -f have the same multipliers (those of the equivalent minimisation)
SGN2	an inequality has a nonnegative multiplier, however it is written (g ≤ 0 / -g ≥ 0, bound / row, path / box)
SGN3	equalities and two-sided constraints carry the signed y = μU − μL
SGN4	the pseudo-Hamiltonian p·f + p⁰ℓ + μ·g (g ≥ 0, μ ≥ 0) is always maximised; p⁰ = −1 for a minimisation, +1 for a maximisation
SGN5	the same constraint has the same dual for every kind (path, box) and every method (direct, indirect); the scale of time-dependent duals is not fixed
SGN6	the multipliers of the NLP solvers are tested in two layers: the native convention of each solver, and the normalised output

Implementation (draft PRs):

• feat(solvers): one multiplier convention for every solver CTSolvers.jl#289: one convention for every NLP solver (option multipliers=:native for the raw output);
• fix(solutions): dual() on missing duals + limits of the path duals at switching times CTModels.jl#483: dual of a box follows SGN2/SGN3, and clear errors for missing duals;
• feat(flows): constraint duals in the Solution of Flow(ocp, law; constraint=:label) CTFlows.jl#494: the duals of a flow solution, from the multiplier given with constraint=:label.

Documentation and cross-package tests: #1015, #1016.