Try debug CaNNoLeS tutorial (#70)

* update CaNNOLeS tutorial

Author: tmigot

tutorials/introduction-to-cannoles/index.jmd

@@ -7,7 +7,9 @@ author: "Abel S. Siqueira"
 CaNNOLeS is a solver for equality-constrained nonlinear least-squares problems, i.e.,
 optimization problems of the form
 
-    min ¹/₂‖F(x)‖²      s. to     c(x) = 0.
+$$
+\min \frac{1}{2}\|F(x)\|^2_2 \quad \text{s.to} \quad c(x) = 0.
+$$
 
 It uses other JuliaSmoothOptimizers packages for development.
 In particular, [NLPModels.jl](https://github.com/JuliaSmoothOptimizers/NLPModels.jl) is used for defining the problem, and [SolverCore](https://github.com/JuliaSmoothOptimizers/SolverCore.jl) for the output.
@@ -24,26 +26,26 @@ Find below a list of the main options of `cannoles`.
 
 ### Tolerances on the problem
 
-| Parameters           | Type          | Default         | Description                                        |
-| -------------------- | ------------- | --------------- | -------------------------------------------------- |
-| ϵtol                 | AbstractFloat | √eps(eltype(x)) | tolerance.                                         |
-| unbounded_threshold  | AbstractFloat | -1e5            | below this threshold the problem is unbounded.     |
-| max_f                | Integer       | 100000          | evaluation limit, e.g. `sum_counters(nls) > max_f` |
-| max_time             | AbstractFloat | 30.             | maximum number of seconds.                         |
-| max_inner            | Integer       | 10000           | maximum number of iterations.                      |
+| Parameters           | Type          | Default           | Description                                        |
+| -------------------- | ------------- | ----------------- | -------------------------------------------------- |
+| ϵtol                 | AbstractFloat | `√eps(eltype(x))` | tolerance                                          |
+| unbounded_threshold  | AbstractFloat | `-1e5`            | below this threshold the problem is unbounded      |
+| max_f                | Integer       | `100000`          | evaluation limit, e.g. `sum_counters(nls) > max_f` |
+| max_time             | AbstractFloat | `30.0`            | maximum number of seconds                          |
+| max_inner            | Integer       | `10000`           | maximum number of iterations                       |
 
 ### Algorithmic parameters
 
-| Parameters                  | Type           | Default           | Description                                        |
-| --------------------------- | -------------- | ----------------- | -------------------------------------------------- |
-| x                           | AbstractVector | copy(nls.meta.x0) | initial guess. |
-| λ                           | AbstractVector | eltype(x)[]       | initial guess for the Lagrange mutlipliers. |
-| method                      | Symbol         | :Newton           | method to compute direction, `:Newton`, `:LM`, `:Newton_noFHess`, or `:Newton_vanishing`. |
-| merit                       | Symbol         | :auglag           | merit function: `:norm1`, `:auglag` |
-| linsolve                    | Symbol         | :ma57             | solver use to compute the factorization: `:ma57`, `:ma97`, `:ldlfactorizations` |
-| check_small_residual        | Bool           | true              | |
-| always_accept_extrapolation | Bool           | false             | |
-| δdec                        | Real           | eltype(x)(0.1)    | |
+| Parameters                  | Type           | Default             | Description                                        |
+| --------------------------- | -------------- | ------------------- | -------------------------------------------------- |
+| x                           | AbstractVector | `nls.meta.x0`       | initial guess |
+| λ                           | AbstractVector | `eltype(x)[]`       | initial guess for the Lagrange mutlipliers |
+| method                      | Symbol         | `:Newton`           | method to compute direction, `:Newton`, `:LM`, `:Newton_noFHess`, or `:Newton_vanishing` |
+| linsolve                    | Symbol         | `:ma57`             | solver use to compute the factorization: `:ma57`, `:ma97`, `:ldlfactorizations` |
+| verbose                     | Int            | `0`                 | if > 0, display iteration details every `verbose` iteration |
+| check_small_residual        | Bool           | `true`              | if `true`, stop whenever $ \|F(x)\|^2_2 \leq $ ``ϵtol`` and $ \|c(x)\|_\infty \leq $ ``√ϵtol`` |
+| always_accept_extrapolation | Bool           | `false`             | if `true`, run even if the extrapolation step fails |
+| δdec                        | Real           | `eltype(x)(0.1)`    | reducing factor on the parameter `δ` |
 
 ## Examples
 
@@ -65,5 +67,5 @@ nls = ADNLSModel(
   [0.0],
   [0.0],
 )
-stats = cannoles(nls, max_time = 10., merit = :auglag)
+stats = cannoles(nls, max_time = 10., linsolve = :ldlfactorizations)
 ```