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Author: mateuszbaran

docs/src/solvers/generalized_cauchy_direction_subsolver.md

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+# Generalized Cauchy Direction subsolver
+
+The Generalized Cauchy Direction (GCD) subsolver is a component in optimization algorithms that handle problems with bound constraints. It solves the following problem
+
+```math
+\begin{align*}
+\operatorname*{arg\,min}_{Y  ∈  T_p D \times \mathcal{M}}&\ m_p(Y) = ⟨X_g, Y⟩_p + \frac{1}{2} ⟨\mathcal{H}_p[Y], Y⟩_p\\
+\text{such that}& \ \exp_p(Y) = \exp_p(\alpha X) \in D \times \mathcal{M} \text{ for some } \alpha \in \mathbb{R}
+\end{align*}
+```
+
+where $X$ is a given direction, the exponential map handles projection of the tangent vector when reaching the boundary, $D$ is a box domain ([`Hyperrectangle`](@extref Manifolds.Hyperrectangle)), $\mathcal{M}$ is a Riemannian manifold, $X_g$ is the gradient of a scalar function $f$ at point $p$ and $\mathcal{H}_p$ is a linear operator that approximates the Hessian of $f$ at $p$.
+
+The solver is currently primarily intended for internal use by optimization algorithms that require bound-constrained subproblem solutions.
+
+## Internal types and method
+
+### Symbols related to the GCP computation
+
+These symbols are directly used by solvers to compute the descent direction corresponding to the Generalized Cauchy point.
+
+```@docs
+Manopt.requires_generalized_cauchy_direction_computation
+Manopt.find_generalized_cauchy_point_direction!
+Manopt.GeneralizedCauchyDirectionFinder
+```
+
+### Symbols related to the Hessian approximation
+
+These symbols are used to evaluate the Hessian approximation at specific tangent vectors during the generalized Cauchy point computation.
+
+```@docs
+Manopt.hessian_value
+Manopt.hessian_value_eb
+```
+
+### Symbols related to bound handling
+
+These are internal symbols used to manage and manipulate bound constraints during the GCP computation.
+
+```@docs
+Manopt.init_updater!
+Manopt.AbstractSegmentHessianUpdater
+Manopt.GenericSegmentHessianUpdater
+Manopt.get_bounds_index
+Manopt.get_stepsize_bound
+Manopt.get_at_bound_index
+Manopt.set_stepsize_bound!
+Manopt.set_zero_at_index!
+```
+
+### Symbols related to specific Hessian approximations
+
+```@docs
+Manopt.LimitedMemorySegmentHessianUpdater
+Manopt.hessian_value_from_inner_products
+Manopt.set_M_current_scale!
+```