conjugate_gradient_descent(M, F, gradF, p=rand(M))
conjugate_gradient_descent(M, gradient_objective, p)

perform a conjugate gradient based descent

\[p_{k+1} = \operatorname{retr}_{p_k} \bigl( s_kδ_k \bigr),\]

where $\operatorname{retr}$ denotes a retraction on the Manifold M and one can employ different rules to update the descent direction $δ_k$ based on the last direction $δ_{k-1}$ and both gradients $\operatorname{grad}f(x_k)$,$\operatorname{grad}f(x_{k-1})$. The Stepsize $s_k$ may be determined by a Linesearch.

Alternatively to f and grad_f you can prodive the AbstractManifoldGradientObjective gradient_objective directly.

Available update rules are SteepestDirectionUpdateRule, which yields a gradient_descent, ConjugateDescentCoefficient (the default), DaiYuanCoefficient, FletcherReevesCoefficient, HagerZhangCoefficient, HestenesStiefelCoefficient, LiuStoreyCoefficient, and PolakRibiereCoefficient. These can all be combined with a ConjugateGradientBealeRestart rule.

They all compute $β_k$ such that this algorithm updates the search direction as

\[\delta_k=\operatorname{grad}f(p_k) + β_k \delta_{k-1}\]

Input

• M : a manifold $\mathcal M$
• f : a cost function $F:\mathcal M→ℝ$ to minimize implemented as a function (M,p) -> v
• grad_f: the gradient $\operatorname{grad}F:\mathcal M → T\mathcal M$ of $F$ implemented also as (M,x) -> X
• p : an initial value $x∈\mathcal M$

Optional

• coefficient : (ConjugateDescentCoefficient <: DirectionUpdateRule) rule to compute the descent direction update coefficient $β_k$, as a functor, i.e. the resulting function maps (amp, cgs, i) -> β, where amp is an AbstractManoptProblem, cgs are the ConjugateGradientDescentState o and i is the current iterate.
• evaluation – (AllocatingEvaluation) specify whether the gradient works by allocation (default) form gradF(M, x) or InplaceEvaluation in place, i.e. is of the form gradF!(M, X, x).
• retraction_method - (default_retraction_method(M, typeof(p))) a retraction method to use.
• stepsize - (ArmijoLinesearch via default_stepsize) A Stepsize function applied to the search direction. The default is a constant step size 1.
• stopping_criterion : (stopWhenAny( stopAtIteration(200), stopGradientNormLess(10.0^-8))) a function indicating when to stop.
• vector_transport_method – (default_vector_transport_method(M, typeof(p))) vector transport method to transport the old descent direction when computing the new descent direction.

If you provide the ManifoldGradientObjective directly, evaluation is ignored.

Output

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

conjugate_gradient_descent!(M, F, gradF, x)
conjugate_gradient_descent!(M, gradient_objective, p; kwargs...)

perform a conjugate gradient based descent in place of x, i.e.

\[p_{k+1} = \operatorname{retr}_{p_k} \bigl( s_k\delta_k \bigr),\]

where $\operatorname{retr}$ denotes a retraction on the Manifold M

Input

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

Alternatively to f and grad_f you can prodive the AbstractManifoldGradientObjective gradient_objective directly.

for more details and options, especially the DirectionUpdateRules, see conjugate_gradient_descent.

ConjugateGradientState <: AbstractGradientSolverState

specify options for a conjugate gradient descent algorithm, that solves a [DefaultManoptProblem].

Fields

• p – the current iterate, a point on a manifold
• X – the current gradient, also denoted as $ξ$ or $X_k$ for the gradient in the $k$th step.
• δ – the current descent direction, i.e. also tangent vector
• β – the current update coefficient rule, see .
• coefficient – (ConjugateDescentCoefficient()) a DirectionUpdateRule function to determine the new β
• stepsize – (default_stepsize(M, ConjugateGradientDescentState; retraction_method=retraction_method)) a Stepsize function
• stop – (StopAfterIteration(500) |StopWhenGradientNormLess(1e-8)) a StoppingCriterion
• retraction_method – (default_retraction_method(M, typeof(p))) a type of retraction
• vector_transport_method – (default_retraction_method(M, typeof(p))) a type of retraction

Constructor

ConjugateGradientState(M, p)

where the last five fields above can be set by their names as keyword and the X can be set to a tangent vector type using the keyword initial_gradient which defaults to zero_vector(M,p), and δ is initialized to a copy of this vector.

See also

conjugate_gradient_descent, DefaultManoptProblem, ArmijoLinesearch

ConjugateGradientBealeRestart <: DirectionUpdateRule

An update rule might require a restart, that is using pure gradient as descent direction, if the last two gradients are nearly orthogonal, cf. Hager, Zhang, Pacific J Optim, 2006, page 12 (in the pdf, 46 in Journal page numbers). This method is named after E. Beale from his proceedings paper in 1972 [Bea72]. This method acts as a decorator to any existing DirectionUpdateRule direction_update.

When obtain from the ConjugateGradientDescentStatecgs the last $p_k,X_k$ and the current $p_{k+1},X_{k+1}$ iterate and the gradient, respectively.

Then a restart is performed, i.e. $β_k = 0$ returned if

\[ \frac{ ⟨X_{k+1}, P_{p_{k+1}\gets p_k}X_k⟩}{\lVert X_k \rVert_{p_k}} > ξ,\]

where $P_{a\gets b}(⋅)$ denotes a vector transport from the tangent space at $a$ to $b$, and $ξ$ is the threshold. The default threshold is chosen as 0.2 as recommended in Powell, Math. Prog., 1977

Constructor

ConjugateGradientBealeRestart(
    direction_update::D,
    threshold=0.2;
    manifold::AbstractManifold = DefaultManifold(),
    vector_transport_method::V=default_vector_transport_method(manifold),
)

ConjugateDescentCoefficient <: DirectionUpdateRule

Computes an update coefficient for the conjugate gradient method, where the ConjugateGradientDescentStatecgds include the last iterates $p_k,X_k$, the current iterates $p_{k+1},X_{k+1}$ of the iterate and the gradient, respectively, and the last update direction $\delta=\delta_k$, based on Fletcher, 1987 adapted to manifolds:

\[β_k = \frac{ \lVert X_{k+1} \rVert_{p_{k+1}}^2 } {\langle -\delta_k,X_k \rangle_{p_k}}.\]

See also conjugate_gradient_descent

Constructor

ConjugateDescentCoefficient(a::StoreStateAction=())

Construct the conjugate descent coefficient update rule, a new storage is created by default.

DaiYuanCoefficient <: DirectionUpdateRule

Computes an update coefficient for the conjugate gradient method, where the ConjugateGradientDescentStatecgds include the last iterates $p_k,X_k$, the current iterates $p_{k+1},X_{k+1}$ of the iterate and the gradient, respectively, and the last update direction $\delta=\delta_k$, based on Dai, Yuan, Siam J Optim, 1999 adapted to manifolds:

Let $\nu_k = X_{k+1} - P_{p_{k+1}\gets p_k}X_k$, where $P_{a\gets b}(⋅)$ denotes a vector transport from the tangent space at $a$ to $b$.

Then the coefficient reads

\[β_k = \frac{ \lVert X_{k+1} \rVert_{p_{k+1}}^2 } {\langle P_{p_{k+1}\gets p_k}\delta_k, \nu_k \rangle_{p_{k+1}}}.\]

See also conjugate_gradient_descent

Constructor

function DaiYuanCoefficient(
    M::AbstractManifold=DefaultManifold(2);
    t::AbstractVectorTransportMethod=default_vector_transport_method(M)
)

Construct the Dai Yuan coefficient update rule, where the parallel transport is the default vector transport and a new storage is created by default.

FletcherReevesCoefficient <: DirectionUpdateRule

Computes an update coefficient for the conjugate gradient method, where the ConjugateGradientDescentStatecgds include the last iterates $p_k,X_k$, the current iterates $p_{k+1},X_{k+1}$ of the iterate and the gradient, respectively, and the last update direction $\delta=\delta_k$, based on Flecther, Reeves, Comput. J, 1964 adapted to manifolds:

\[β_k = \frac{\lVert X_{k+1}\rVert_{p_{k+1}}^2}{\lVert X_k\rVert_{x_{k}}^2}.\]

See also conjugate_gradient_descent

Constructor

FletcherReevesCoefficient(a::StoreStateAction=())

Construct the Fletcher Reeves coefficient update rule, a new storage is created by default.

HagerZhangCoefficient <: DirectionUpdateRule

Computes an update coefficient for the conjugate gradient method, where the ConjugateGradientDescentStatecgds include the last iterates $p_k,X_k$, the current iterates $p_{k+1},X_{k+1}$ of the iterate and the gradient, respectively, and the last update direction $\delta=\delta_k$, based on Hager, Zhang, SIAM J Optim, 2005. adapted to manifolds: let $\nu_k = X_{k+1} - P_{p_{k+1}\gets p_k}X_k$, where $P_{a\gets b}(⋅)$ denotes a vector transport from the tangent space at $a$ to $b$.

\[β_k = \Bigl\langle\nu_k - \frac{ 2\lVert \nu_k\rVert_{p_{k+1}}^2 }{ \langle P_{p_{k+1}\gets p_k}\delta_k, \nu_k \rangle_{p_{k+1}} } P_{p_{k+1}\gets p_k}\delta_k, \frac{X_{k+1}}{ \langle P_{p_{k+1}\gets p_k}\delta_k, \nu_k \rangle_{p_{k+1}} } \Bigr\rangle_{p_{k+1}}.\]

This method includes a numerical stability proposed by those authors.

See also conjugate_gradient_descent

Constructor

function HagerZhangCoefficient(t::AbstractVectorTransportMethod)
function HagerZhangCoefficient(M::AbstractManifold = DefaultManifold(2))

Construct the Hager Zhang coefficient update rule, where the parallel transport is the default vector transport and a new storage is created by default.

HestenesStiefelCoefficient <: DirectionUpdateRule

Computes an update coefficient for the conjugate gradient method, where the ConjugateGradientDescentStatecgds include the last iterates $p_k,X_k$, the current iterates $p_{k+1},X_{k+1}$ of the iterate and the gradient, respectively, and the last update direction $\delta=\delta_k$, based on Heestenes, Stiefel, J. Research Nat. Bur. Standards, 1952 adapted to manifolds as follows:

Let $\nu_k = X_{k+1} - P_{p_{k+1}\gets p_k}X_k$. Then the update reads

\[β_k = \frac{\langle X_{k+1}, \nu_k \rangle_{p_{k+1}} } { \langle P_{p_{k+1}\gets p_k} \delta_k, \nu_k\rangle_{p_{k+1}} },\]

where $P_{a\gets b}(⋅)$ denotes a vector transport from the tangent space at $a$ to $b$.

Constructor

function HestenesStiefelCoefficient(transport_method::AbstractVectorTransportMethod)
function HestenesStiefelCoefficient(M::AbstractManifold = DefaultManifold(2))

Construct the Heestens Stiefel coefficient update rule, where the parallel transport is the default vector transport and a new storage is created by default.

See also conjugate_gradient_descent

LiuStoreyCoefficient <: DirectionUpdateRule

Computes an update coefficient for the conjugate gradient method, where the ConjugateGradientDescentStatecgds include the last iterates $p_k,X_k$, the current iterates $p_{k+1},X_{k+1}$ of the iterate and the gradient, respectively, and the last update direction $\delta=\delta_k$, based on Lui, Storey, J. Optim. Theoru Appl., 1991 adapted to manifolds:

Let $\nu_k = X_{k+1} - P_{p_{k+1}\gets p_k}X_k$, where $P_{a\gets b}(⋅)$ denotes a vector transport from the tangent space at $a$ to $b$.

Then the coefficient reads

\[β_k = - \frac{ \langle X_{k+1},\nu_k \rangle_{p_{k+1}} } {\langle \delta_k,X_k \rangle_{p_k}}.\]

See also conjugate_gradient_descent

Constructor

function LiuStoreyCoefficient(t::AbstractVectorTransportMethod)
function LiuStoreyCoefficient(M::AbstractManifold = DefaultManifold(2))

Construct the Lui Storey coefficient update rule, where the parallel transport is the default vector transport and a new storage is created by default.

PolakRibiereCoefficient <: DirectionUpdateRule

Computes an update coefficient for the conjugate gradient method, where the ConjugateGradientDescentStatecgds include the last iterates $p_k,X_k$, the current iterates $p_{k+1},X_{k+1}$ of the iterate and the gradient, respectively, and the last update direction $\delta=\delta_k$, based on Poliak, Ribiere, ESIAM Math. Modelling Num. Anal., 1969 and Polyak, USSR Comp. Math. Math. Phys., 1969 adapted to manifolds:

Let $\nu_k = X_{k+1} - P_{p_{k+1}\gets p_k}X_k$, where $P_{a\gets b}(⋅)$ denotes a vector transport from the tangent space at $a$ to $b$.

Then the update reads

\[β_k = \frac{ \langle X_{k+1}, \nu_k \rangle_{p_{k+1}} } {\lVert X_k \rVert_{p_k}^2 }.\]

Constructor

function PolakRibiereCoefficient(
    M::AbstractManifold=DefaultManifold(2);
    t::AbstractVectorTransportMethod=default_vector_transport_method(M)
)

Construct the PolakRibiere coefficient update rule, where the parallel transport is the default vector transport and a new storage is created by default.

See also conjugate_gradient_descent

SteepestDirectionUpdateRule <: DirectionUpdateRule

The simplest rule to update is to have no influence of the last direction and hence return an update $β = 0$ for all ConjugateGradientDescentStatecgds

See also conjugate_gradient_descent