Riemannian quasiNewton methods
Manopt.quasi_Newton
— Functionquasi_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
 Compute the search direction $η_k = \mathcal{B}_k [\operatorname{grad}f (p_k)]$ or solve $\mathcal{H}_k [η_k] = \operatorname{grad}f (p_k)]$.
 Determine a suitable stepsize $α_k$ along the curve $\gamma(α) = R_{p_k}(α η_k)$ e.g. by using
WolfePowellLinesearch
.  Compute
p_{k+1} = R_{p_k}(α_k η_k)
`.  Define $s_k = T_{p_k, α_k η_k}(α_k η_k)$ and $y_k = \operatorname{grad}f(p_{k+1})  T_{p_k, α_k η_k}(\operatorname{grad}f(p_k))$.
 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 space(s) to represent the Hessian (inverse).cautious_update
– (false
) – whether or not to use aQuasiNewtonCautiousDirectionUpdate
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) formgradF(M, x)
orInplaceEvaluation
in place, i.e. is of the formgradF!(M, X, x)
.initial_operator
– (Matrix{Float64}(I,n,n)
) initial matrix to use die the approximation, wheren=manifold_dimension(M)
, see alsoscale_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 representationretraction_method
– (default_retraction_method(M, typeof(p))
) a retraction method to use, by default the exponential map.scale_initial_operator
 (true
) scale initial operator with $\frac{⟨s_k,y_k⟩_{p_k}}{\lVert y_k\rVert_{p_k}}$ in the computationstabilize
– (true
) stabilize the method numerically by projecting computed (Newton) directions to the tangent space to reduce numerical errorsstepsize
– (WolfePowellLinesearch
(retraction_method, vector_transport_method)
) specify aStepsize
.stopping_criterion
 (StopWhenAny(StopAfterIteration(max(1000, memory_size)), StopWhenGradientNormLess(10^(6))
) specify aStoppingCriterion
vector_transport_method
– (default_vector_transport_method(M, typeof(p))
) a vector transport to use.
Output
the obtained (approximate) minimizer $p^*$, see get_solver_return
for details.
Manopt.quasi_Newton!
— Functionquasi_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 asgradF(M,p)
.x
– an initial value $x ∈ \mathcal{M}$.
For all optional parameters, see quasi_Newton
.
Background
The aim is to minimize a realvalued function on a Riemannian manifold, i.e.
\[\min f(x), \quad x ∈ \mathcal{M}.\]
Riemannian quasiNewtonian methods are as generalizations of their Euclidean counterparts Riemannian line search methods. These methods determine a search direction $η_k ∈ T_{x_k} \mathcal{M}$ at the current iterate $x_k$ and a suitable stepsize $α_k$ along $\gamma(α) = R_{x_k}(α η_k)$, where $R: T \mathcal{M} →\mathcal{M}$ is a retraction. The next iterate is obtained by
\[x_{k+1} = R_{x_k}(α_k η_k).\]
In quasiNewton methods, the search direction is given by
\[η_k = {\mathcal{H}_k}^{1}[\operatorname{grad}f (x_k)] = \mathcal{B}_k [\operatorname{grad} (x_k)],\]
where $\mathcal{H}_k : T_{x_k} \mathcal{M} →T_{x_k} \mathcal{M}$ is a positive definite selfadjoint operator, which approximates the action of the Hessian $\operatorname{Hess} f (x_k)[⋅]$ and $\mathcal{B}_k = {\mathcal{H}_k}^{1}$. The idea of quasiNewton methods is instead of creating a complete new approximation of the Hessian operator $\operatorname{Hess} f(x_{k+1})$ or its inverse at every iteration, the previous operator $\mathcal{H}_k$ or $\mathcal{B}_k$ is updated by a convenient formula using the obtained information about the curvature of the objective function during the iteration. The resulting operator $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$ acts on the tangent space $T_{x_{k+1}} \mathcal{M}$ of the freshly computed iterate $x_{k+1}$. In order to get a welldefined method, the following requirements are placed on the new operator $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$ that is created by an update. Since the Hessian $\operatorname{Hess} f(x_{k+1})$ is a selfadjoint operator on the tangent space $T_{x_{k+1}} \mathcal{M}$, and $\mathcal{H}_{k+1}$ approximates it, we require that $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$ is also selfadjoint on $T_{x_{k+1}} \mathcal{M}$. In order to achieve a steady descent, we want $η_k$ to be a descent direction in each iteration. Therefore we require, that $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$ is a positive definite operator on $T_{x_{k+1}} \mathcal{M}$. In order to get information about the curvature of the objective function into the new operator $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$, we require that it satisfies a form of a Riemannian quasiNewton equation:
\[\mathcal{H}_{k+1} [T_{x_k \rightarrow x_{k+1}}({R_{x_k}}^{1}(x_{k+1}))] = \operatorname{grad}(x_{k+1})  T_{x_k \rightarrow x_{k+1}}(\operatorname{grad}f(x_k))\]
or
\[\mathcal{B}_{k+1} [\operatorname{grad}f(x_{k+1})  T_{x_k \rightarrow x_{k+1}}(\operatorname{grad}f(x_k))] = T_{x_k \rightarrow x_{k+1}}({R_{x_k}}^{1}(x_{k+1}))\]
where $T_{x_k \rightarrow x_{k+1}} : T_{x_k} \mathcal{M} →T_{x_{k+1}} \mathcal{M}$ and the chosen retraction $R$ is the associated retraction of $T$. We note that, of course, not all updates in all situations will meet these conditions in every iteration. For specific quasiNewton updates, the fulfillment of the Riemannian curvature condition, which requires that
\[g_{x_{k+1}}(s_k, y_k) > 0\]
holds, is a requirement for the inheritance of the selfadjointness and positive definiteness of the $\mathcal{H}_k$ or $\mathcal{B}_k$ to the operator $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$. Unfortunately, the fulfillment of the Riemannian curvature condition is not given by a step size $\alpha_k > 0$ that satisfies the generalized Wolfe conditions. However, in order to create a positive definite operator $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$ in each iteration, the socalled locking condition was introduced in Huang, Gallican, Absil, SIAM J. Optim., 2015, which requires that the isometric vector transport $T^S$, which is used in the update formula, and its associate retraction $R$ fulfill
\[T^{S}{x, ξ_x}(ξ_x) = β T^{R}{x, ξ_x}(ξ_x), \quad β = \frac{\lVert ξ_x \rVert_x}{\lVert T^{R}{x, ξ_x}(ξ_x) \rVert_{R_{x}(ξ_x)}},\]
where $T^R$ is the vector transport by differentiated retraction. With the requirement that the isometric vector transport $T^S$ and its associated retraction $R$ satisfies the locking condition and using the tangent vector
\[y_k = {β_k}^{1} \operatorname{grad}f(x_{k+1})  T^{S}{x_k, α_k η_k}(\operatorname{grad}f(x_k)),\]
where
\[β_k = \frac{\lVert α_k η_k \rVert_{x_k}}{\lVert T^{R}{x_k, α_k η_k}(α_k η_k) \rVert_{x_{k+1}}},\]
in the update, it can be shown that choosing a stepsize $α_k > 0$ that satisfies the Riemannian Wolfe conditions leads to the fulfillment of the Riemannian curvature condition, which in turn implies that the operator generated by the updates is positive definite. In the following we denote the specific operators in matrix notation and hence use $H_k$ and $B_k$, respectively.
Direction Updates
In general there are different ways to compute a fixed AbstractQuasiNewtonUpdateRule
. In general these are represented by
Manopt.AbstractQuasiNewtonDirectionUpdate
— TypeAbstractQuasiNewtonDirectionUpdate
An abstract representation of an Quasi Newton Update rule to determine the next direction given current QuasiNewtonState
.
All subtypes should be functors, i.e. one should be able to call them as H(M,x,d)
to compute a new direction update.
Manopt.QuasiNewtonMatrixDirectionUpdate
— TypeQuasiNewtonMatrixDirectionUpdate <: AbstractQuasiNewtonDirectionUpdate
These AbstractQuasiNewtonDirectionUpdate
s represent any quasiNewton 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, i.e.
\[\text{Solve} \quad \hat{η_k} =  H_k \widehat{\operatorname{grad}f(x_k)}\]
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 matrixvector multiplication, i.e.
\[\hat{η_k} =  B_k \widehat{\operatorname{grad}f(x_k)}\]
where $B_k$ is the matrix representing the operator with respect to the basis $\{b_i\}^{n}_{i=1}$ and $\widehat{\operatorname{grad}f(x_k)}$ as above. 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 quasiNewton 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.
Fields
update
– aAbstractQuasiNewtonUpdateRule
.basis
– the basis.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.vector_transport_method
– (vector_transport_method
)anAbstractVectorTransportMethod
Constructor
QuasiNewtonMatrixDirectionUpdate(M::AbstractManifold, update, basis, matrix;
scale=true, vector_transport_method=default_vector_transport_method(M))
Generate the Update rule with defaults from a manifold and the names corresponding to the fields above.
See also
QuasiNewtonLimitedMemoryDirectionUpdate
QuasiNewtonCautiousDirectionUpdate
AbstractQuasiNewtonDirectionUpdate
Manopt.QuasiNewtonLimitedMemoryDirectionUpdate
— TypeQuasiNewtonLimitedMemoryDirectionUpdate <: AbstractQuasiNewtonDirectionUpdate
This AbstractQuasiNewtonDirectionUpdate
represents the limitedmemory Riemanian BFGS update, where the approximating operator is represented by $m$ stored pairs of tangent vectors $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=km}^{k1}$ in the $k$th iteration. For the calculation of the search direction $η_k$, the generalisation of the twoloop recursion is used (see Huang, Gallican, Absil, SIAM J. Optim., 2015), 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 $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=km}^{k1}$, the gradient $\operatorname{grad}f(x_k)$ of the objective function $f$ in $x_k$ and the positive definite selfadjoint operator
\[\mathcal{B}^{(0)}_k[⋅] = \frac{g_{x_k}(s_{k1}, y_{k1})}{g_{x_k}(y_{k1}, y_{k1})} \; \mathrm{id}_{T_{x_k} \mathcal{M}}[⋅]\]
are used. The twoloop 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 $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=km}^{k1}$, 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, i.e. $k < m$, the previously stored vector pairs $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=km}^{k1}$ 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 $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=km+1}^{k1}$ are transported into the tangent space $T_{x_{k+1}} \mathcal{M}$. After that we calculate and store $s_k = \widetilde{s}_k = T^{S}_{x_k, α_k η_k}(α_k η_k)$ and $y_k = \widetilde{y}_k$. This process ensures that new information about the objective function is always included and the old, probably no longer relevant, information is discarded.
Fields
memory_s
– the set of the stored (and transported) search directions times step size $\{ \widetilde{s}_i\}_{i=km}^{k1}$.memory_y
– set of the stored gradient differences $\{ \widetilde{y}_i\}_{i=km}^{k1}$.ξ
– a variable used in the twoloop recursion.ρ
– a variable used in the twoloop recursion.scale
–vector_transport_method
– aAbstractVectorTransportMethod
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=1.0
project=true
)
See also
InverseBFGS
QuasiNewtonCautiousDirectionUpdate
AbstractQuasiNewtonDirectionUpdate
Manopt.QuasiNewtonCautiousDirectionUpdate
— TypeQuasiNewtonCautiousDirectionUpdate <: AbstractQuasiNewtonDirectionUpdate
These AbstractQuasiNewtonDirectionUpdate
s represent any quasiNewton update rule, which are based on the idea of a socalled cautious update. The search direction is calculated as given in QuasiNewtonMatrixDirectionUpdate
or QuasiNewtonLimitedMemoryDirectionUpdate
, butut the update then is only executed if
\[\frac{g_{x_{k+1}}(y_k,s_k)}{\lVert s_k \rVert^{2}_{x_{k+1}}} \geq \theta(\lVert \operatorname{grad}f(x_k) \rVert_{x_k}),\]
is satisfied, where $\theta$ is a monotone increasing function satisfying $\theta(0) = 0$ and $\theta$ is strictly increasing at $0$. If this is not the case, the corresponding update will be 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 $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=km}^{k1}$ 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 Huang, Absil, Gallivan, SIAM J. Optim., 2018, taking into account that the corresponding step size is chosen.
Fields
update
– anAbstractQuasiNewtonDirectionUpdate
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 memorz based update rule.
See also
QuasiNewtonMatrixDirectionUpdate
QuasiNewtonLimitedMemoryDirectionUpdate
Hessian Update Rules
Using
Manopt.update_hessian!
— Functionupdate_hessian!(d, amp, st, p_old, iter)
update the hessian wihtin the QuasiNewtonState
o
given a AbstractManoptProblem
amp
as well as the an AbstractQuasiNewtonDirectionUpdate
d
and the last iterate p_old
. Note that the current (iter
th) iterate is already stored in o.x
.
See also AbstractQuasiNewtonUpdateRule
for the different rules that are available within d
.
the following update formulae for either $H_{k+1}$ or $B_{k+1}$ are available.
Manopt.AbstractQuasiNewtonUpdateRule
— TypeAbstractQuasiNewtonUpdateRule
Specify a type for the different AbstractQuasiNewtonDirectionUpdate
s, that is, e.g. for a QuasiNewtonMatrixDirectionUpdate
there are several differeent updates to the matrix, while the default for QuasiNewtonLimitedMemoryDirectionUpdate
the most prominent is InverseBFGS
.
Manopt.BFGS
— TypeBFGS <: AbstractQuasiNewtonUpdateRule
indicates in AbstractQuasiNewtonDirectionUpdate
that the Riemanian BFGS update is used in the Riemannian quasiNewton method.
We 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.
Manopt.DFP
— TypeDFP <: AbstractQuasiNewtonUpdateRule
indicates in an AbstractQuasiNewtonDirectionUpdate
that the Riemanian DFP update is used in the Riemannian quasiNewton method.
We 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.
Manopt.Broyden
— TypeBroyden <: AbstractQuasiNewtonUpdateRule
indicates in AbstractQuasiNewtonDirectionUpdate
that the Riemanian Broyden update is used in the Riemannian quasiNewton method, which is as a convex combination of BFGS
and DFP
.
We 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)
Manopt.SR1
— TypeSR1 <: AbstractQuasiNewtonUpdateRule
indicates in AbstractQuasiNewtonDirectionUpdate
that the Riemanian SR1 update is used in the Riemannian quasiNewton method.
We 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 Nocedal, Wright, Springer, 2006.
Constructor
SR1(r::Float64=1.0)
Generate the SR1
update, which by default does not include the check (since the default sets $t<0$`)
Manopt.InverseBFGS
— TypeInverseBFGS <: AbstractQuasiNewtonUpdateRule
indicates in AbstractQuasiNewtonDirectionUpdate
that the inverse Riemanian BFGS update is used in the Riemannian quasiNewton method.
We 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.
Manopt.InverseDFP
— TypeInverseDFP <: AbstractQuasiNewtonUpdateRule
indicates in AbstractQuasiNewtonDirectionUpdate
that the inverse Riemanian DFP update is used in the Riemannian quasiNewton method.
We 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.
Manopt.InverseBroyden
— TypeInverseBroyden <: AbstractQuasiNewtonUpdateRule
Indicates in AbstractQuasiNewtonDirectionUpdate
that the Riemanian Broyden update is used in the Riemannian quasiNewton method, which is as a convex combination of InverseBFGS
and InverseDFP
.
We 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)
Manopt.InverseSR1
— TypeInverseSR1 <: AbstractQuasiNewtonUpdateRule
indicates in AbstractQuasiNewtonDirectionUpdate
that the inverse Riemanian SR1 update is used in the Riemannian quasiNewton method.
We 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 Nocedal, Wright, Springer, 2006.
Constructor
InverseSR1(r::Float64=1.0)
Generate the InverseSR1
update, which by default does not include the check, since the default sets $t<0$`.
State
The quasi Newton algorithm is based on a DefaultManoptProblem
.
Manopt.QuasiNewtonState
— TypeQuasiNewtonState <: AbstractManoptSolverState
These Quasi Newton AbstractManoptSolverState
represent any quasiNewton based method and can be used with any update rule for the direction.
Fields
p
– the current iterate, a point on a manifoldX
– the current gradientsk
– the current stepyk
the current gradient differencedirection_update
 anAbstractQuasiNewtonDirectionUpdate
rule.retraction_method
– anAbstractRetractionMethod
stop
– aStoppingCriterion
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(1e6),
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
Literature
 [HAG18]

W. Huang, P.A. Absil and K. A. Gallivan. A Riemannian BFGS method without differentiated retraction for nonconvex optimization problems. SIAM Journal on Optimization 28, 470–495 (2018).
 [HGA15]

W. Huang, K. A. Gallivan and P.A. Absil. A Broyden class of quasiNewton methods for Riemannian optimization. SIAM Journal on Optimization 25, 1660–1685 (2015).
 [NW06]

J. Nocedal and S. J. Wright. Numerical Optimization. Springer, New York (2006).