Expand documentation on convergence options

Fixes #1102

docs/src/user/config.md

@@ -35,12 +35,15 @@ Special methods for bounded univariate optimization:
 * `Brent()`
 * `GoldenSection()`
 
-### General Options
+### [General Options](@id config-general)
+
 In addition to the solver, you can alter the behavior of the Optim package by using the following keywords:
 
-* `x_tol`: Absolute tolerance in changes of the input vector `x`, in infinity norm. Defaults to `0.0`.
-* `f_tol`: Relative tolerance in changes of the objective value. Defaults to `0.0`.
-* `g_tol`: Absolute tolerance in the gradient. If `g_tol` is a scalar (the default), convergence is achieved when `norm(g, Inf) ≤ g_tol`; if `g_tol` is supplied as a vector, then each component must satisfy `abs(g[i]) ≤ g_tol[i]`. Defaults to `1e-8`. For gradient-free methods (e.g., Nelder-Meade), this gets re-purposed to control the main convergence tolerance in a solver-specific manner.
+* `x_tol` (alternatively, `x_abstol`): Absolute tolerance in changes of the input vector `x`, in infinity norm. Concretely, if `|x-x'| ≤ x_tol` on successive evaluation points `x` and `x'`, convergence is achieved. Defaults to `0.0`.
+* `x_reltol`: Relative tolerance in changes of the input vector `x`, in infinity norm. Concretely, if `|x-x'| ≤ x_reltol * |x|`, convergence is achieved. Defaults to `0.0`
+* `f_tol` (alternatively, `f_reltol`): Relative tolerance in changes of the objective value. Defaults to `0.0`.
+* `f_abstol`: Absolute tolerance in changes of the objective value. Defaults to `0.0`.
+* `g_tol` (alternatively, `g_abstol`): Absolute tolerance in the gradient. If `g_tol` is a scalar (the default), convergence is achieved when `norm(g, Inf) ≤ g_tol`; if `g_tol` is supplied as a vector, then each component must satisfy `abs(g[i]) ≤ g_tol[i]`. Defaults to `1e-8`. For gradient-free methods (e.g., Nelder-Meade), this gets re-purposed to control the main convergence tolerance in a solver-specific manner.
 * `f_calls_limit`: A soft upper limit on the number of objective calls. Defaults to `0` (unlimited).
 * `g_calls_limit`: A soft upper limit on the number of gradient calls. Defaults to `0` (unlimited).
 * `h_calls_limit`: A soft upper limit on the number of Hessian calls. Defaults to `0` (unlimited).

docs/src/user/minimization.md

@@ -31,6 +31,12 @@ For better performance and greater precision, you can pass your own gradient fun
 optimize(f, x0, LBFGS(); autodiff = :forward)
 ```
 
+!!! note
+    For most real-world problems, you may want to carefully consider the appropriate convergence criteria.
+    By default, algorithms that support gradients converge if `|g| ≤ 1e-8`. Depending on how your variables are scaled,
+    this may or may not be appropriate. See [configuration](@ref config-general) for more information about your options.
+    Examining traces (`Options(show_trace=true)`) during optimization may provide insight about when convergence is achieved in practice.
+
 For the Rosenbrock example, the analytical gradient can be shown to be:
 ```jl
 function g!(G, x)
@@ -65,7 +71,7 @@ Now we can use Newton's method for optimization by running:
 ```jl
 optimize(f, g!, h!, x0)
 ```
-Which defaults to `Newton()` since a Hessian function was provided. Like gradients, the Hessian function will be ignored if you use a method that does not require it:
+which defaults to `Newton()` since a Hessian function was provided. Like gradients, the Hessian function will be ignored if you use a method that does not require it:
 ```jl
 optimize(f, g!, h!, x0, LBFGS())
 ```
@@ -74,6 +80,12 @@ because of the potentially low accuracy of approximations to the Hessians. Other
 than Newton's method, none of the algorithms provided by the Optim package employ
 exact Hessians.
 
+As a reminder, it's advised to set your convergence criteria manually based on
+your knowledge of the problem:
+```
+optimize(f, g!, h!, x0, Optim.Options(g_tol = 1e-12))
+```
+
 ## Box Constrained Optimization
 
 A primal interior-point algorithm for simple "box" constraints (lower and upper bounds) is available. Reusing our Rosenbrock example from above, boxed minimization is performed as follows: