Documentation for nonlinear least squares (NLS) problems (#411)

* Clarify documentation for nonlinear least squares (NLS) problems: classification, implementation, and usage for contributors and users.

* addressing review comments

* Apply suggestions from code review

Co-authored-by: Tangi Migot <tangi.migot@gmail.com>

* Remove NLS guidelines from contributing.md

Removed guidelines for Nonlinear Least Squares (NLS) problems from the contributing documentation.

* addressing review comments

---------

Co-authored-by: Tangi Migot <tangi.migot@gmail.com>

Author: arnavk23

docs/src/meta.md

@@ -17,7 +17,7 @@ OptimizationProblems.AMPGO02_meta
 ```
 See `? OptimizationProblems.meta` for more documentation on the various entries and their default values.
 
-This structre is completed by getters to access the number of variables, `get_nameoftheproblem_nvar`,the number of constraints, `get_nameoftheproblem_ncon`, the number of linear constraints, `get_nameoftheproblem_nlin`, the number of nonlinear constraints, `get_nameoftheproblem_nnln`, the number of equality constraints, `get_nameoftheproblem_nequ`, and the number of inequality constraints, `get_nameoftheproblem_nineq`.
+This structure is completed by getters to access the number of variables, `get_nameoftheproblem_nvar`, the number of constraints, `get_nameoftheproblem_ncon`, the number of linear constraints, `get_nameoftheproblem_nlin`, the number of nonlinear constraints, `get_nameoftheproblem_nnln`, the number of equality constraints, `get_nameoftheproblem_nequ`, and the number of inequality constraints, `get_nameoftheproblem_nineq`.
 ```@example 1
 OptimizationProblems.get_AMPGO02_nvar()
 ```
@@ -50,3 +50,13 @@ adproblems = (
   eval(Meta.parse("ADNLPProblems.$(pb[:name])()")) for pb in eachrow(names_pb_vars)
 )
 ```
+
+### Nonlinear Least Squares Problems (NLS) 
+Problems with `:objtype` set to `:least_squares` are nonlinear least squares problems (NLS). For these, you can access the number of NLS equations using a getter like `get_nameoftheproblem_nls_nequ()`.
+```@example 1
+OptimizationProblems.get_lanczos1_nls_nequ()
+```
+To filter all NLS problems in the metadata DataFrame:
+```@example 1
+nls_problems = OptimizationProblems.meta[OptimizationProblems.meta.objtype .== :least_squares, :name]
+```

docs/src/tutorial.md

@@ -47,7 +47,7 @@ using OptimizationProblems, OptimizationProblems.ADNLPProblems
 problems = OptimizationProblems.meta[!, :name]
 length(problems)
 ```
-Similarly, to the PureJuMP models, it suffices to select any of this problem to get the model.
+Similarly, to the PureJuMP models, it suffices to select any of these problems to get the model.
 ``` @example ex2
 nlp = zangwil3()
 ```
@@ -62,8 +62,31 @@ One of the advantages of these problems is that they are type-stable. Indeed, on
 ``` @example ex2
 nlp16_12 = woods(n=12, type=Float16)
 ```
-Then, all the API will be compatible with the precised type.
+Then, all the API will be compatible with the specified type.
 ``` @example ex2
 using NLPModels
 obj(nlp16_12, zeros(Float16, 12))
 ```
+
+### Nonlinear Least Squares Problems (NLS)
+
+Some problems are classified as nonlinear least squares (NLS). These problems may provide both an `ADNLPModel` and an `ADNLSModel` implementation, and dispatch to one or the other using the `use_nls` keyword argument, see [`arglina`](https://github.com/JuliaSmoothOptimizers/OptimizationProblems.jl/blob/main/src/ADNLPProblems/arglina.jl) for a concrete example of this pattern.
+
+Using `ADNLSModel` can be preferable when a solver exploits the least-squares structure directly (residual vector and Jacobian), which is often more efficient and numerically robust than treating the same problem as a generic nonlinear program (`ADNLPModel`).
+
+To obtain the least squares model (`ADNLSModel`), use the `use_nls=true` keyword argument when constructing the problem:
+``` @example ex2
+lanczos1_nlp = lanczos1()
+lanczos1_nls = lanczos1(use_nls=true)
+typeof(lanczos1_nlp)
+typeof(lanczos1_nls)
+```
+
+To list all NLS problems:
+``` @example ex2
+nls_problems = OptimizationProblems.meta[OptimizationProblems.meta.objtype .== :least_squares, :name]
+```
+To access the number of NLS equations for a problem:
+``` @example ex2
+OptimizationProblems.get_lanczos1_nls_nequ()
+```