Fix typos and improve clarity in paper.md

Author: MohamedLaghdafHABIBOULLAH

paper/paper.md

@@ -46,15 +46,15 @@ The library provides a modular and extensible framework for experimenting with n
 - **Levenbergh-Marquardt methods (LM, LMTR)** [@aravkin-baraldi-orban-2024].
 - **Augmented Lagrangian methods (AL)** [@demarchi-jia-kanzow-mehlitz-2023].
 
-These methods rely on the gradient and optionnally on the Hessian(-vector) information of the smooth part $f$ and the proximal mapping of the nonsmooth part $h$ in order to compute steps.
+These methods rely on the gradient and optionally on the Hessian(-vector) information of the smooth part $f$ and the proximal mapping of the nonsmooth part $h$ in order to compute steps.
 Then, the objective function $f + h$ is used only to accept or reject trial points.
 Moreover, they can handle cases where Hessian approximations are unbounded [@diouane-habiboullah-orban-2024;@leconte-orban-2023-2], making the package particularly suited for large-scale, ill-conditioned, and nonsmooth problems.
 
 # Statement of need
 
 ## Model-based framework for nonsmooth methods
 
-In Julia, \eqref{eq:nlp} can be solved using [ProximalAlgorithms.jl](https://github.com/JuliaFirstOrder/ProximalAlgorithms.jl), which implements in-place, first-order, line-search–based methods[@stella-themelis-sopasakis-patrinos-2017;@themelis-stella-patrinos-2017].
+In Julia, \eqref{eq:nlp} can be solved using [ProximalAlgorithms.jl](https://github.com/JuliaFirstOrder/ProximalAlgorithms.jl), which implements in-place, first-order, line-search–based methods [@stella-themelis-sopasakis-patrinos-2017;@themelis-stella-patrinos-2017].
 Most of these methods are generally splitting schemes that alternate between taking steps along the gradient of the smooth part $f$ (or quasi-Newton directions) and applying proximal steps on the nonsmooth part $h$.
 Currently, [ProximalAlgorithms.jl](https://github.com/JuliaFirstOrder/ProximalAlgorithms.jl) provides only L-BFGS as a quasi-Newton option.
 By contrast, [RegularizedOptimization.jl](https://github.com/JuliaSmoothOptimizers/RegularizedOptimization.jl) focuses on model-based approaches such as trust-region and quadratic regularization algorithms.
@@ -69,15 +69,17 @@ On the one hand, smooth problems $f$ can be defined via [NLPModels.jl](https://g
 Large collections of such problems are available in [Cutest.jl](https://github.com/JuliaSmoothOptimizers/CUTEst.jl) [@orban-siqueira-cutest-2020] and [OptimizationProblems.jl](https://github.com/JuliaSmoothOptimizers/OptimizationProblems.jl) [@migot-orban-siqueira-optimizationproblems-2023].
 Another option is to use [RegularizedProblems.jl](https://github.com/JuliaSmoothOptimizers/RegularizedProblems.jl), which provides problem instances commonly used in the nonsmooth optimization literature.
 
-On the other hand, Hessian approximations of these functions, including quasi-Newton and diagonal schemes, can be specified through [LinearOperators.jl](https://github.com/JuliaSmoothOptimizers/LinearOperators.jl), which represents Hessians as linear operators and implements efficient Hessian–vector products.
+On the other hand, nonsmooth terms $h$ can be modeled using [ProximalOperators.jl](https://github.com/JuliaSmoothOptimizers/ProximalOperators.jl), which provides a broad collection of nonsmooth functions, together with [ShiftedProximalOperators.jl](https://github.com/JuliaSmoothOptimizers/ShiftedProximalOperators.jl), which provides shifted proximal mappings for nonsmooth functions.
 
-Finally, nonsmooth terms $h$ can be modeled using [ProximalOperators.jl](https://github.com/JuliaSmoothOptimizers/ProximalOperators.jl), which provides a broad collection of nonsmooth functions, together with [ShiftedProximalOperators.jl](https://github.com/JuliaSmoothOptimizers/ShiftedProximalOperators.jl), which provides shifted proximal mappings for nonsmooth functions.
+## Support for Hessians and Hessian approximations of the smooth part $f$
 
-## Support for Hessians of the smooth part $f$
+In contrast to [ProximalAlgorithms.jl](https://github.com/JuliaFirstOrder/ProximalAlgorithms.jl), [RegularizedOptimization.jl](https://github.com/JuliaSmoothOptimizers/RegularizedOptimization.jl) methods such as **R2N** and **TR** methods such as R2N and TR support exact Hessians as well as several Hessian approximations of $f$, which can significantly improve convergence rates—especially for ill-conditioned problems.
 
-In contrast to [ProximalAlgorithms.jl](https://github.com/JuliaFirstOrder/ProximalAlgorithms.jl), [RegularizedOptimization.jl](https://github.com/JuliaSmoothOptimizers/RegularizedOptimization.jl) methods such as **R2N** and **TR** support Hessians of $f$, which can significantly improve convergence rates, especially for ill-conditioned problems.
-Hessians can be obtained via automatic differentiation through [ADNLPModels.jl](https://github.com/JuliaSmoothOptimizers/ADNLPModels.jl) or supplied directly as Hessian–vector products $v \mapsto Hv$.
-This enables algorithms to exploit second-order information without explicitly forming dense (or sparse) Hessians, which is often prohibitively expensive in both computation and memory, particularly in high-dimensional settings.
+Exact Hessians can be obtained via automatic differentiation through [ADNLPModels.jl](https://github.com/JuliaSmoothOptimizers/ADNLPModels.jl) or supplied directly as Hessian–vector products $v \mapsto Hv$.
+
+Hessian approximations (e.g., quasi-Newton and diagonal schemes) can be specified via [LinearOperators.jl](https://github.com/JuliaSmoothOptimizers/LinearOperators.jl), which represents Hessians as linear operators and provides efficient implementations of Hessian–vector products.
+
+This design allows algorithms to exploit second-order information **without** explicitly forming dense (or even sparse) Hessian matrices, which is often prohibitively expensive in time and memory, particularly at large scale.
 
 ## Requirements of the RegularizedProblems.jl
 
@@ -115,7 +117,7 @@ Extensive documentation is provided, including a user guide, API reference, and
 Aqua.jl is used to test the package dependencies.
 Documentation is built using Documenter.jl.
 
-## Solvers caracteristics
+## Solvers characteristics
 
 All solvers in [RegularizedOptimization.jl](https://github.com/JuliaSmoothOptimizers/RegularizedOptimization.jl) are implemented in an in-place fashion, minimizing memory allocations during the resolution process.
 Moreover, they implement non-monotone strategies to accept trial points, which can enhance algorithmic performance in practice [@leconte-orban-2023;@diouane-habiboullah-orban-2024].
@@ -144,10 +146,8 @@ We compare the performance of our solvers with (**PANOC**) solver [@stella-theme
 We illustrate the capabilities of [RegularizedOptimization.jl](https://github.com/JuliaSmoothOptimizers/RegularizedOptimization.jl) on two nonsmooth and nonconvex problems:
 
 - **Support Vector Machine (SVM) with $\ell^{1/2}$ penalty** for image classification [@aravkin-baraldi-orban-2024].  
-- **Nonnegative Matrix Factorization (NNMF) with $\ell_0$ penalty and constraints** [@kim-park-2008].
+- **Nonnegative Matrix Factorization (NNMF) with $\ell_0$ penalty and bound constraints** [@kim-park-2008].
 
-Both problems are of the form $\min f(x) + h(x)$ with $f$ nonconvex and $h$ nonsmooth.  
-The NNMF problem can be set up in a similar way to the SVM case, with $h$ given by an $\ell_0$ norm and additional nonnegativity constraints.
 Below is a condensed example showing how to define and solve such problems:
 
 ```julia
@@ -164,7 +164,7 @@ stats  = RegularizedExecutionStats(reg_nlp)
 solve!(solver, reg_nlp, stats; x=f.meta.x0, atol=1e-4, rtol=1e-4, verbose=0, sub_kwargs=(max_iter=200,))
 solve!(solver, reg_nlp, stats; x=f.meta.x0, atol=1e-5, rtol=1e-5, verbose=0, sub_kwargs=(max_iter=200,))
 ```
-The NNMF problem can be set up in a similar way, replacing the model by nnmf_model(...) with bound constraints, $h$ by an $\ell_0$ norm and use an L-BFGS Hessian approximation.
+The NNMF problem can be set up in a similar way, replacing the model by nnmf_model(...), $h$ by an $\ell_0$ norm and use an L-BFGS Hessian approximation.
 
 ### Numerical results