What does "Solved to an acceptable level" mean in the context of NonlinearModelFit in M12

Question

I made a simple cosine fit to some data:

NonlinearModelFit[
                    Data[[1;;]] /. a_Around :> a["Value"],
                    {
                        A Cos[ω x + ϕ] + c,
                        {-π <= ϕ <= +π}
                    }, 
                    {{A, -0.0017}, {ω, 400}, {c, 0.01}, {ϕ}}, x,
                    VarianceEstimatorFunction->(1&), 
                    Weights -> (1/(Data[[1;;,2]] /. a_Around :> a["Uncertainty"])^2),
                    ConfidenceLevel->0.95, MaxIterations->10000
                ];

I can provide the data if necessary. The fit seems to converge successfully and the plotted result makes sense, along with the resultant fit parameters. However I do get the error message/warning:

NonlinearModelFit::acceptlev: Solved to acceptable level.

Firstly, what does this mean? How can I remove this (improve my fitting routine)? Does it have any impact on the actual results on the fit.

I have one suspicion that it is because the fit is converging near a fit constraint boundary.

I'm using Mathematica 12, so possibly why some users haven't seen this error message before and also why I couldn't find any results when I searched the error message.

The data:

{{33322.984, Around[0.007816405492811665,  0.00027463653231123993]},
 {33322.988, Around[0.010073344779686584,  0.0007117756195902295]}, 
 {33322.991, Around[0.010108725016292291,  0.0009945416303385364]}, 
 {33322.992, Around[0.008967820799845302,  0.0013022059402730695]}, 
 {33322.994, Around[0.00802865952165009,   0.0012786303668924815]}, 
 {33322.997, Around[0.0068887219740959,    0.0007474404408052571]}, 
 {33323.,    Around[0.008130289312875589,  0.00022863757915000962]}, 
 {33323.003, Around[0.009914276423570985,  0.00030355873713380484]}, 
 {33323.006, Around[0.010214891502463226,  0.001117808525551174]}, 
 {33323.008, Around[0.009013144046862024,  0.0011652145970101099]}, 
 {33323.009, Around[0.008126883089375795,  0.0013667571907391915]}, 
 {33323.012, Around[0.0068149354267696815, 0.0009386542257803183]}, 
 {33323.016, Around[0.008337596520600767,  0.00020742323701885597]}}

Answer 1

It would be nice if there would be some control over the "acceptable-tol" parameter in IPOPT, which as @Lukas points out is the method being used in the OP as well as here. The "tol" parameter seems to be set to the maximum of PrecisionGoal and AccuracyGoal, but the "acceptable-tol" parameter is left at whatever value is the default.

The OP's example did not produce the warning message for me in V12.3, so I adapted an example so that it produces the message from Even though the fit seems correct, NonlinearModelFit throws a failed convergence error.

data = {{2, -2.99380668585}, {4, -2.99413053462}, {6, -2.99439488497}, {8, 
-2.99467836024}, {10, -2.99491958936}, {12, -2.99519218472}, {14, 
-2.99538900867}, {16, -2.99562768004}, {18, -2.99584876062}, {20, 
-2.99601713877}, {22, -2.99619549077}, {24, -2.99637350562}};

accTol = 1*^-12; (* does not produce NonlinearModelFit::acceptlev *)
accTol = 1*^-8;  (* produces NonlinearModelFit::acceptlev *)
WithCleanup[
 ipoptopts = Options[IPOPTLink`Private`IPOPTSolve]
 ; Options[IPOPTLink`Private`IPOPTSolve] = 
  Append[ipoptopts, "acceptable_tol" -> N@accTol]
 ,
 fit[accTol] = 
  NonlinearModelFit[
   data, {-a + ρ r^m Cos[m ϕ + ψ], 
    r > 0}, {a, ρ, ϕ, ψ, r}, m, MaxIterations -> 100, 
   PrecisionGoal -> 12, Method -> "IPOPT"]
 ,
 Options[IPOPTLink`Private`IPOPTSolve] = ipoptopts
 ]

NonlinearModelFit::acceptlev: Solved to acceptable level.

Here we can see it makes a difference, from which I infer that option is working. (Run the above code twice, once with each setting, accTol = 1*^-8 and accTol = 1*^-12.)

fit[1*^-8]@"BestFitParameters"
fit[1*^-12]@"BestFitParameters"
(*
{a -> 2.99862, ρ -> -0.358466, ϕ -> -47.1236, ψ -> -4.69809, r -> 0.952692}
{a -> 2.99857, ρ -> -0.358328, ϕ -> -47.1236, ψ -> -4.69824, r -> 0.951456}
*) 

Answer 2

Note that this is NOT an answer to the question but rather a critique of the model and data considered.

I get the same "acceptable" message with Mathematica 12.3 but there is a simple approach to avoid that message. There are really only 2 significant figures for the predictor variable. Subtracting 33322.98 from all predictor values gets one a more numerically stable result and avoids the "acceptable" message:

(* Original model *)
nlm = NonlinearModelFit[Data[[1 ;;]] /. a_Around :> a["Value"],
   {A Cos[ω x + ϕ] + c, {-π <= ϕ <= +π}}, {{A, -0.0017}, {ω, 400}, {c, 0.01}, {ϕ}}, x, 
   VarianceEstimatorFunction -> (1 &), 
   Weights -> (1/(Data[[1 ;;, 2]] /. a_Around :> a["Uncertainty"])^2),
    ConfidenceLevel -> 0.95, MaxIterations -> 10000];

(* Modified model *)
nlm2 = NonlinearModelFit[Data[[1 ;;]] /. a_Around :> a["Value"], 
  {A Cos[ω (x - 33322.98) + ϕ] + c, {-π <= ϕ <= +π}}, {{A, -0.0017}, {ω, 400}, {c, 0.01}, {ϕ}}, x,
  VarianceEstimatorFunction -> (1 &), 
  Weights -> (1/(Data[[1 ;;, 2]] /. a_Around :> a["Uncertainty"])^2),
  ConfidenceLevel -> 0.95, MaxIterations -> 10000];

nlm["CorrelationMatrix"] // MatrixForm

nlm2["CorrelationMatrix"] // MatrixForm

Both models give the same predictions (other than maybe very small round-off errors) but the modified model has a much better behaved parameter correlation matrix.

But!!! Looking at the fit of either model suggests that the precision values are way overestimated (i.e., they're no good).

Show[ListPlot[Data], Plot[nlm2[x], {x, Min[Data[[All, 1]]], Max[Data[[All, 1]]]}]]

The data points appear much closer to the fitted curve than any of the precision values would suggest (and I'm assuming that the precision values are standard errors associated with each observation).

Fitting without weights gets one a standard error of estimate much smaller than any of the precision values. Some appears wrong with the precision values or I've made some incorrect assumptions about the precision values.

nlm3 = NonlinearModelFit[Data[[1 ;;]] /. a_Around :> a["Value"], {A Cos[ω (x - 33322.98) + ϕ] +  c,
  {-π <= ϕ <= +π}}, {{A, -0.0017}, {ω, 400}, {c, 0.01}, {ϕ}}, x,
  ConfidenceLevel -> 0.95, MaxIterations -> 10000];
nlm3["EstimatedVariance"]^0.5
(* 0.000163099 *)