Riemannian quasi-Newton methods

quasi_Newton(M, f, grad_f, p)

Perform a quasi Newton iteration for f on the manifold M starting in the point p.

The $k$th iteration consists of

1. Compute the search direction $η_k = -\mathcal{B}_k [\operatorname{grad}f (p_k)]$ or solve $\mathcal{H}_k [η_k] = -\operatorname{grad}f (p_k)]$.
2. Determine a suitable stepsize $α_k$ along the curve $\gamma(α) = R_{p_k}(α η_k)$, usually by using WolfePowellLinesearch.
3. Compute p_{k+1} = R_{p_k}(α_k η_k)`.
4. Define $s_k = T_{p_k, α_k η_k}(α_k η_k)$ and $y_k = \operatorname{grad}f(p_{k+1}) - T_{p_k, α_k η_k}(\operatorname{grad}f(p_k))$.
5. Compute the new approximate Hessian $H_{k+1}$ or its inverse $B_k$.

Input

• M a manifold $\mathcal{M}$.
• f a cost function $F : \mathcal{M} →ℝ$ to minimize.
• grad_f the gradient $\operatorname{grad}F : \mathcal{M} →T_x\mathcal M$ of $F$.
• p an initial value $p ∈ \mathcal{M}$.

Optional

• basis: (DefaultOrthonormalBasis()) basis within the tangent spaces

to represent the Hessian (inverse).

• cautious_update: (false) whether or not to use a QuasiNewtonCautiousDirectionUpdate
• cautious_function: ((x) -> x*10^(-4)) a monotone increasing function that is zero at 0 and strictly increasing at 0 for the cautious update.
• direction_update: (InverseBFGS()) the update rule to use.
• evaluation: (AllocatingEvaluation) specify whether the gradient works by allocation (default) form gradF(M, x) or InplaceEvaluation in place of form gradF!(M, X, x).
• initial_operator: (Matrix{Float64}(I,n,n)) initial matrix to use die the approximation, where n=manifold_dimension(M), see also scale_initial_operator.
• memory_size: (20) limited memory, number of $s_k, y_k$ to store. Set to a negative value to use a full memory representation
• retraction_method: (default_retraction_method(M, typeof(p))) a retraction method to use
• scale_initial_operator: (true) scale initial operator with $\frac{⟨s_k,y_k⟩_{p_k}}{\lVert y_k\rVert_{p_k}}$ in the computation
• stabilize: (true) stabilize the method numerically by projecting computed (Newton-) directions to the tangent space to reduce numerical errors
• stepsize: (WolfePowellLinesearch(retraction_method, vector_transport_method)) specify a Stepsize.
• stopping_criterion: (StopAfterIteration(max(1000, memory_size)) |StopWhenGradientNormLess(1e-6)) specify a StoppingCriterion
• vector_transport_method: (default_vector_transport_method(M, typeof(p))) a vector transport to use.
• nondescent_direction_behavior: (:step_towards_negative_gradient) specify how non-descent direction is handled. This can be
  • `:step_towards_negative_gradient – the direction is replaced with negative gradient, a message is stored.
  • :ignore – the check is not performed, so any computed direction is accepted. No message is stored.
  • any other value performs the check, keeps the direction but stores a message.
  A stored message can be displayed using DebugMessages.

Output

the obtained (approximate) minimizer $p^*$, see get_solver_return for details.

quasi_Newton!(M, F, gradF, x; options...)

Perform a quasi Newton iteration for F on the manifold M starting in the point x using a retraction $R$ and a vector transport $T$.

Input

• M a manifold $\mathcal{M}$.
• F a cost function $F: \mathcal{M} →ℝ$ to minimize.
• gradF the gradient $\operatorname{grad}F : \mathcal{M} → T_x\mathcal M$ of $F$ implemented as gradF(M,p).
• x an initial value $x ∈ \mathcal{M}$.

For all optional parameters, see quasi_Newton.

AbstractQuasiNewtonDirectionUpdate

An abstract representation of an Quasi Newton Update rule to determine the next direction given current QuasiNewtonState.

All subtypes should be functors, they should be callable as H(M,x,d) to compute a new direction update.

QuasiNewtonMatrixDirectionUpdate <: AbstractQuasiNewtonDirectionUpdate

The QuasiNewtonMatrixDirectionUpdate represent a quasi-Newton update rule, where the operator is stored as a matrix. A distinction is made between the update of the approximation of the Hessian, $H_k \mapsto H_{k+1}$, and the update of the approximation of the Hessian inverse, $B_k \mapsto B_{k+1}$. For the first case, the coordinates of the search direction $η_k$ with respect to a basis $\{b_i\}^{n}_{i=1}$ are determined by solving a linear system of equations

\[\text{Solve} \quad \hat{η_k} = - H_k \widehat{\operatorname{grad}f(x_k)},\]

where $H_k$ is the matrix representing the operator with respect to the basis $\{b_i\}^{n}_{i=1}$ and $\widehat{\operatorname{grad}f(x_k)}$ represents the coordinates of the gradient of the objective function $f$ in $x_k$ with respect to the basis $\{b_i\}^{n}_{i=1}$. If a method is chosen where Hessian inverse is approximated, the coordinates of the search direction $η_k$ with respect to a basis $\{b_i\}^{n}_{i=1}$ are obtained simply by matrix-vector multiplication

\[\hat{η_k} = - B_k \widehat{\operatorname{grad}f(x_k)},\]

where $B_k$ is the matrix representing the operator with respect to the basis $\{b_i\}^{n}_{i=1}$ and $\widehat{\operatorname{grad}f(x_k)}$. In the end, the search direction $η_k$ is generated from the coordinates $\hat{eta_k}$ and the vectors of the basis $\{b_i\}^{n}_{i=1}$ in both variants. The AbstractQuasiNewtonUpdateRule indicates which quasi-Newton update rule is used. In all of them, the Euclidean update formula is used to generate the matrix $H_{k+1}$ and $B_{k+1}$, and the basis $\{b_i\}^{n}_{i=1}$ is transported into the upcoming tangent space $T_{x_{k+1}} \mathcal{M}$, preferably with an isometric vector transport, or generated there.

Provided functors

• (mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction
• (η, mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction in-place of η

Fields

• basis: an AbstractBasis to use in the tangent spaces
• matrix: (Matrix{Float64}(I, manifold_dimension(M), manifold_dimension(M))) the matrix which represents the approximating operator.
• scale: (`true) indicates whether the initial matrix (= identity matrix) should be scaled before the first update.
• update: a AbstractQuasiNewtonUpdateRule.
• vector_transport_method: (vector_transport_method)an AbstractVectorTransportMethod

Constructor

QuasiNewtonMatrixDirectionUpdate(
    M::AbstractManifold,
    update,
    basis::B=DefaultOrthonormalBasis(),
    m=Matrix{Float64}(I, manifold_dimension(M), manifold_dimension(M));
    kwargs...
)

Keyword arguments

• scale, vector_transport_method for the two fields

Generate the Update rule with defaults from a manifold and the names corresponding to the fields.

See also

QuasiNewtonLimitedMemoryDirectionUpdate QuasiNewtonCautiousDirectionUpdate AbstractQuasiNewtonDirectionUpdate

QuasiNewtonLimitedMemoryDirectionUpdate <: AbstractQuasiNewtonDirectionUpdate

This AbstractQuasiNewtonDirectionUpdate represents the limited-memory Riemannian BFGS update, where the approximating operator is represented by $m$ stored pairs of tangent vectors $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=k-m}^{k-1}$ in the $k$-th iteration. For the calculation of the search direction $η_k$, the generalisation of the two-loop recursion is used (see [HGA15]), since it only requires inner products and linear combinations of tangent vectors in $T_{x_k} \mathcal{M}$. For that the stored pairs of tangent vectors $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=k-m}^{k-1}$, the gradient $\operatorname{grad}f(x_k)$ of the objective function $f$ in $x_k$ and the positive definite self-adjoint operator

\[\mathcal{B}^{(0)}_k[⋅] = \frac{g_{x_k}(s_{k-1}, y_{k-1})}{g_{x_k}(y_{k-1}, y_{k-1})} \; \mathrm{id}_{T_{x_k} \mathcal{M}}[⋅]\]

are used. The two-loop recursion can be understood as that the InverseBFGS update is executed $m$ times in a row on $\mathcal{B}^{(0)}_k[⋅]$ using the tangent vectors $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=k-m}^{k-1}$, and in the same time the resulting operator $\mathcal{B}^{LRBFGS}_k [⋅]$ is directly applied on $\operatorname{grad}f(x_k)$. When updating there are two cases: if there is still free memory, $k < m$, the previously stored vector pairs $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=k-m}^{k-1}$ have to be transported into the upcoming tangent space $T_{x_{k+1}} \mathcal{M}$. If there is no free memory, the oldest pair $\{ \widetilde{s}_{k−m}, \widetilde{y}_{k−m}\}$ has to be discarded and then all the remaining vector pairs $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=k-m+1}^{k-1}$ are transported into the tangent space $T_{x_{k+1}} \mathcal{M}$. After that the new values $s_k = \widetilde{s}_k = T^{S}_{x_k, α_k η_k}(α_k η_k)$ and $y_k = \widetilde{y}_k$ are stored at the beginning. This process ensures that new information about the objective function is always included and the old, probably no longer relevant, information is discarded.

Provided functors

• (mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction
• (η, mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction in-place of η

Fields

• memory_s the set of the stored (and transported) search directions times step size $\{ \widetilde{s}_i\}_{i=k-m}^{k-1}$.
• memory_y set of the stored gradient differences $\{ \widetilde{y}_i\}_{i=k-m}^{k-1}$.
• ξ a variable used in the two-loop recursion.
• ρ a variable used in the two-loop recursion.
• scale initial scaling of the Hessian
• vector_transport_method a AbstractVectorTransportMethod
• message a string containing a potential warning that might have appeared

Constructor

QuasiNewtonLimitedMemoryDirectionUpdate(
    M::AbstractManifold,
    x,
    update::AbstractQuasiNewtonUpdateRule,
    memory_size;
    initial_vector=zero_vector(M,x),
    scale::Real=1.0
    project::Bool=true
)

See also

InverseBFGS QuasiNewtonCautiousDirectionUpdate AbstractQuasiNewtonDirectionUpdate

QuasiNewtonCautiousDirectionUpdate <: AbstractQuasiNewtonDirectionUpdate

These AbstractQuasiNewtonDirectionUpdates represent any quasi-Newton update rule, which are based on the idea of a so-called cautious update. The search direction is calculated as given in QuasiNewtonMatrixDirectionUpdate or QuasiNewtonLimitedMemoryDirectionUpdate, butut the update then is only executed if

\[\frac{g_{x_{k+1}}(y_k,s_k)}{\lVert s_k \rVert^{2}_{x_{k+1}}} ≥ θ(\lVert \operatorname{grad}f(x_k) \rVert_{x_k}),\]

is satisfied, where $θ$ is a monotone increasing function satisfying $θ(0) = 0$ and $θ$ is strictly increasing at $0$. If this is not the case, the corresponding update is skipped, which means that for QuasiNewtonMatrixDirectionUpdate the matrix $H_k$ or $B_k$ is not updated. The basis $\{b_i\}^{n}_{i=1}$ is nevertheless transported into the upcoming tangent space $T_{x_{k+1}} \mathcal{M}$, and for QuasiNewtonLimitedMemoryDirectionUpdate neither the oldest vector pair $\{ \widetilde{s}_{k−m}, \widetilde{y}_{k−m}\}$ is discarded nor the newest vector pair $\{ \widetilde{s}_{k}, \widetilde{y}_{k}\}$ is added into storage, but all stored vector pairs $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=k-m}^{k-1}$ are transported into the tangent space $T_{x_{k+1}} \mathcal{M}$. If InverseBFGS or InverseBFGS is chosen as update, then the resulting method follows the method of [HAG18], taking into account that the corresponding step size is chosen.

Provided functors

• (mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction
• (η, mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction in-place of η

Fields

• update: an AbstractQuasiNewtonDirectionUpdate
• θ: a monotone increasing function satisfying $θ(0) = 0$ and $θ$ is strictly increasing at $0$.

Constructor

QuasiNewtonCautiousDirectionUpdate(U::QuasiNewtonMatrixDirectionUpdate; θ = x -> x)
QuasiNewtonCautiousDirectionUpdate(U::QuasiNewtonLimitedMemoryDirectionUpdate; θ = x -> x)

Generate a cautious update for either a matrix based or a limited memory based update rule.

See also

QuasiNewtonMatrixDirectionUpdate QuasiNewtonLimitedMemoryDirectionUpdate

update_hessian!(d, amp, st, p_old, iter)

update the Hessian within the QuasiNewtonState o given a AbstractManoptProblem amp as well as the an AbstractQuasiNewtonDirectionUpdate d and the last iterate p_old. Note that the current (iterth) iterate is already stored in o.x.

See also AbstractQuasiNewtonUpdateRule for the different rules that are available within d.

AbstractQuasiNewtonUpdateRule

Specify a type for the different AbstractQuasiNewtonDirectionUpdates, that is for a QuasiNewtonMatrixDirectionUpdate there are several different updates to the matrix, while the default for QuasiNewtonLimitedMemoryDirectionUpdate the most prominent is InverseBFGS.

BFGS <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the Riemannian BFGS update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{H}_k^\mathrm{BFGS}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[H^\mathrm{BFGS}_{k+1} = \widetilde{H}^\mathrm{BFGS}_k + \frac{y_k y^{\mathrm{T}}_k }{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{H}^\mathrm{BFGS}_k s_k s^{\mathrm{T}}_k \widetilde{H}^\mathrm{BFGS}_k }{s^{\mathrm{T}}_k \widetilde{H}^\mathrm{BFGS}_k s_k}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

DFP <: AbstractQuasiNewtonUpdateRule

indicates in an AbstractQuasiNewtonDirectionUpdate that the Riemannian DFP update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{H}_k^\mathrm{DFP}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[H^\mathrm{DFP}_{k+1} = \Bigl( \mathrm{id}_{T_{x_{k+1}} \mathcal{M}} - \frac{y_k s^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k} \Bigr) \widetilde{H}^\mathrm{DFP}_k \Bigl( \mathrm{id}_{T_{x_{k+1}} \mathcal{M}} - \frac{s_k y^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k} \Bigr) + \frac{y_k y^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

Broyden <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the Riemannian Broyden update is used in the Riemannian quasi-Newton method, which is as a convex combination of BFGS and DFP.

Denote by $\widetilde{H}_k^\mathrm{Br}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[H^\mathrm{Br}_{k+1} = \widetilde{H}^\mathrm{Br}_k - \frac{\widetilde{H}^\mathrm{Br}_k s_k s^{\mathrm{T}}_k \widetilde{H}^\mathrm{Br}_k}{s^{\mathrm{T}}_k \widetilde{H}^\mathrm{Br}_k s_k} + \frac{y_k y^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k} + φ_k s^{\mathrm{T}}_k \widetilde{H}^\mathrm{Br}_k s_k \Bigl( \frac{y_k}{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{H}^\mathrm{Br}_k s_k}{s^{\mathrm{T}}_k \widetilde{H}^\mathrm{Br}_k s_k} \Bigr) \Bigl( \frac{y_k}{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{H}^\mathrm{Br}_k s_k}{s^{\mathrm{T}}_k \widetilde{H}^\mathrm{Br}_k s_k} \Bigr)^{\mathrm{T}}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively, and $φ_k$ is the Broyden factor which is :constant by default but can also be set to :Davidon.

Constructor

Broyden(φ, update_rule::Symbol = :constant)

SR1 <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the Riemannian SR1 update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{H}_k^\mathrm{SR1}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[H^\mathrm{SR1}_{k+1} = \widetilde{H}^\mathrm{SR1}_k + \frac{ (y_k - \widetilde{H}^\mathrm{SR1}_k s_k) (y_k - \widetilde{H}^\mathrm{SR1}_k s_k)^{\mathrm{T}} }{ (y_k - \widetilde{H}^\mathrm{SR1}_k s_k)^{\mathrm{T}} s_k }\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

This method can be stabilized by only performing the update if denominator is larger than $r\lVert s_k\rVert_{x_{k+1}}\lVert y_k - \widetilde{H}^\mathrm{SR1}_k s_k \rVert_{x_{k+1}}$ for some $r>0$. For more details, see Section 6.2 in [NW06].

Constructor

SR1(r::Float64=-1.0)

Generate the SR1 update.

InverseBFGS <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the inverse Riemannian BFGS update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{B}_k^\mathrm{BFGS}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[B^\mathrm{BFGS}_{k+1} = \Bigl( \mathrm{id}_{T_{x_{k+1}} \mathcal{M}} - \frac{s_k y^{\mathrm{T}}_k }{s^{\mathrm{T}}_k y_k} \Bigr) \widetilde{B}^\mathrm{BFGS}_k \Bigl( \mathrm{id}_{T_{x_{k+1}} \mathcal{M}} - \frac{y_k s^{\mathrm{T}}_k }{s^{\mathrm{T}}_k y_k} \Bigr) + \frac{s_k s^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

InverseDFP <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the inverse Riemannian DFP update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{B}_k^\mathrm{DFP}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[B^\mathrm{DFP}_{k+1} = \widetilde{B}^\mathrm{DFP}_k + \frac{s_k s^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{B}^\mathrm{DFP}_k y_k y^{\mathrm{T}}_k \widetilde{B}^\mathrm{DFP}_k}{y^{\mathrm{T}}_k \widetilde{B}^\mathrm{DFP}_k y_k}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

InverseBroyden <: AbstractQuasiNewtonUpdateRule

Indicates in AbstractQuasiNewtonDirectionUpdate that the Riemannian Broyden update is used in the Riemannian quasi-Newton method, which is as a convex combination of InverseBFGS and InverseDFP.

Denote by $\widetilde{H}_k^\mathrm{Br}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[B^\mathrm{Br}_{k+1} = \widetilde{B}^\mathrm{Br}_k - \frac{\widetilde{B}^\mathrm{Br}_k y_k y^{\mathrm{T}}_k \widetilde{B}^\mathrm{Br}_k}{y^{\mathrm{T}}_k \widetilde{B}^\mathrm{Br}_k y_k} + \frac{s_k s^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k} + φ_k y^{\mathrm{T}}_k \widetilde{B}^\mathrm{Br}_k y_k \Bigl( \frac{s_k}{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{B}^\mathrm{Br}_k y_k}{y^{\mathrm{T}}_k \widetilde{B}^\mathrm{Br}_k y_k} \Bigr) \Bigl( \frac{s_k}{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{B}^\mathrm{Br}_k y_k}{y^{\mathrm{T}}_k \widetilde{B}^\mathrm{Br}_k y_k} \Bigr)^{\mathrm{T}}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively, and $φ_k$ is the Broyden factor which is :constant by default but can also be set to :Davidon.

Constructor

InverseBroyden(φ, update_rule::Symbol = :constant)

InverseSR1 <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the inverse Riemannian SR1 update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{B}_k^\mathrm{SR1}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[B^\mathrm{SR1}_{k+1} = \widetilde{B}^\mathrm{SR1}_k + \frac{ (s_k - \widetilde{B}^\mathrm{SR1}_k y_k) (s_k - \widetilde{B}^\mathrm{SR1}_k y_k)^{\mathrm{T}} }{ (s_k - \widetilde{B}^\mathrm{SR1}_k y_k)^{\mathrm{T}} y_k }\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

This method can be stabilized by only performing the update if denominator is larger than $r\lVert y_k\rVert_{x_{k+1}}\lVert s_k - \widetilde{H}^\mathrm{SR1}_k y_k \rVert_{x_{k+1}}$ for some $r>0$. For more details, see Section 6.2 in [NW06].

Constructor

InverseSR1(r::Float64=-1.0)

Generate the InverseSR1.

QuasiNewtonState <: AbstractManoptSolverState

These Quasi Newton AbstractManoptSolverState represent any quasi-Newton based method and can be used with any update rule for the direction.

Fields

• p the current iterate, a point on a manifold
• X the current gradient
• sk the current step
• yk the current gradient difference
• direction_update an AbstractQuasiNewtonDirectionUpdate rule.
• nondescent_direction_behavior a Symbol to specify how to handle direction that are not descent ones.
• retraction_method an AbstractRetractionMethod
• stepsize a Stepsize
• stop a StoppingCriterion

as well as for internal use

• p_old the last iterate
• η the current update direction
• X_old the last gradient
• nondescent_direction_value the value from the last inner product check for descent directions

Constructor

QuasiNewtonState(
    M::AbstractManifold,
    x;
    initial_vector=zero_vector(M,x),
    direction_update::D=QuasiNewtonLimitedMemoryDirectionUpdate(M, x, InverseBFGS(), 20;
        vector_transport_method=vector_transport_method,
    )
    stopping_criterion=StopAfterIteration(1000) | StopWhenGradientNormLess(1e-6),
    retraction_method::RM=default_retraction_method(M, typeof(p)),
    vector_transport_method::VTM=default_vector_transport_method(M, typeof(p)),
    stepsize=default_stepsize(M; QuasiNewtonState)
)

See also

quasi_Newton