# Problems

A problem usually contains its cost function and provides an implementation to access the cost

Manopt.ProblemType
Problem{T}

Describe the problem that should be optimized by stating all properties, that do not change during an optimization or that are dependent of a certain solver.

The parameter T can be used to distinguish problems with different representations or implementations. The default parameter AllocatingEvaluation, which might be slower but easier to use. The usually faster parameter value is MutatingEvaluation

See Options for the changing and solver dependent properties.

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A problem can be of different type, more specifically, whether its containing functions, for example to compute the gradient, work with allocation or without it. To be precise, an allocation function X = gradF(x) allocates memory for its result X, while gradF!(X,x) does not.

## Cost based problem

Manopt.CostProblemType
CostProblem{T, Manifold, TCost} <: Problem{T}

speficy a problem for solvers just based on cost functions, i.e. gradient free ones.

Fields

• M – a manifold $\mathcal M$
• cost – a function $F: \mathcal M → ℝ$ to minimize

Constructors

CostProblem(M, cost; evaluation=AllocatingEvaluation())

Generate a problem. While this Problem does not have any allocating functions, the type T can be set for consistency reasons with other problems.

NelderMead

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Manopt.GradientProblemType
GradientProblem{T} <: AbstractGradientProblem{T}

specify a problem for gradient based algorithms.

Fields

• M – a manifold $\mathcal M$
• cost – a function $F: \mathcal M → ℝ$ to minimize
• gradient!! – the gradient $\operatorname{grad}F:\mathcal M → \mathcal T\mathcal M$ of the cost function $F$.

Depending on the AbstractEvaluationType T the gradient has to be provided

Constructors

GradientProblem(M, cost, gradient; evaluation=AllocatingEvaluation())

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Manopt.StochasticGradientProblemType
StochasticGradientProblem <: Problem

A stochastic gradient problem consists of

• a Manifold M
• a(n optional) cost function f(x) = \displaystyle\sum{i=1}^n fi(x)
• an array of gradients, i.e. a function that returns and array or an array of functions

$\{\operatorname{grad}f_i\}_{i=1}^n$, where both variants can be given in the allocating variant and the array of function may also be provided as mutating functions (X,x) -> X.

Constructors

StochasticGradientProblem(M::AbstractManifold, gradF::Function;
cost=Missing(), evaluation=AllocatingEvaluation()
)
cost=Missing(), evaluation=AllocatingEvaluation()
)

Create a Stochastic gradient problem with an optional cost and the gradient either as one function (returning an array) or a vector of functions.

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Manopt.get_gradientFunction
get_gradient(O::Options)

return the (last stored) gradient within OptionsO. By default also undecorates the options beforehand

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get_gradient(p::AbstractGradientProblem{T},x)
get_gradient!(p::AbstractGradientProblem{T}, X, x)

evaluate the gradient of a AbstractGradientProblem{T}p at the point x.

The evaluation is done in place of X for the !-variant. The T=AllocatingEvaluation problem might still allocate memory within. When the non-mutating variant is called with a T=MutatingEvaluation memory for the result is allocated.

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get_gradient(p::StochasticGradientProblem, k, x)
get_gradient!(p::StochasticGradientProblem, Y, k, x)

Evaluate one of the summands gradients $\operatorname{grad}f_k$, $k∈\{1,…,n\}$, at x (in place of Y).

Note that for the MutatingEvaluation based problem and a single function for the stochastic gradient mutating variant is not available, since it would require too many allocations.

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get_gradient(P::AlternatingGradientProblem, x)
get_gradient!(P::AlternatingGradientProblem, Y, x)

Evaluate all summands gradients at a point x on the ProductManifold M (in place of Y)

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get_gradient(p::AlternatingGradientProblem, k, x)
get_gradient!(p::AlternatingGradientProblem, Y, k, x)

Evaluate one of the component gradients $\operatorname{grad}f_k$, $k∈\{1,…,n\}$, at x (in place of Y).

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Manopt.get_gradientsFunction
get_gradients(P::StochasticGradientProblem, x)
get_gradients!(P::StochasticGradientProblem, Y, x)

Evaluate all summands gradients $\{\operatorname{grad}f_i\}_{i=1}^n$ at x (in place of Y).

Note that for the MutatingEvaluation based problem and a single function for the stochastic gradient, the allocating variant is not available.

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Manopt.SubGradientProblemType
SubGradientProblem <: Problem

Fields

• manifold – a Manifold
• cost – the function $F$ to be minimized
• subgradient – a function returning a subgradient $\partial F$ of $F$

Constructor

SubGradientProblem(M, f, ∂f)

Generate the [Problem] for a subgradient problem, i.e. a function f on the manifold M and a function ∂f that returns an element from the subdifferential at a point.

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Manopt.get_subgradientFunction
get_subgradient(p, q)
get_subgradient!(p, X, q)

Evaluate the (sub)gradient of a SubGradientProblemp at the point q.

The evaluation is done in place of X for the !-variant. The T=AllocatingEvaluation problem might still allocate memory within. When the non-mutating variant is called with a T=MutatingEvaluation memory for the result is allocated.

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## Constrained based problem

Manopt.ConstrainedProblemType
ConstrainedProblem{T, Manifold} <: AbstractGradientProblem{T}

Describes the constrained problem

\begin{aligned} \operatorname*{arg\,min}_{p ∈\mathcal{M}} & f(p)\\ \text{subject to } &g_i(p)\leq0 \quad \text{ for all } i=1,…,m,\\ \quad &h_j(p)=0 \quad \text{ for all } j=1,…,n. \end{aligned}

It consists of

• an AbstractManifold M
• an cost function $f(p)$
• the gradient of $f$, $\operatorname{grad}f(p)$ (cf. AbstractGradientProblem)
• inequality constraints $g(p)$, either a function g returning a vector or a vector [g1, g2,...,gm] of functions.
• equality constraints $h(p)$, either a function h returning a vector or a vector [h1, h2,...,hn] of functions.
• gradient(s) of the inequality constraints $\operatorname{grad}g(p) ∈ (T_p\mathcal M)^m$, either a function or a vector of functions.
• gradient(s) of the equality constraints $\operatorname{grad}h(p) ∈ (T_p\mathcal M)^n$, either a function or a vector of functions.

There are two ways to specify the constraints $g$ and $h$.

1. as one Function returning a vector in $\mathbb R^m$ and $\mathbb R^n$ respecively. This might be easier to implement but requires evaluating all constraints even if only one is needed.
2. as a AbstractVector{<:Function} where each function returns a real number. This requires each constrant to be implemented as a single function, but it is possible to evaluate also only a single constraint.

The gradients $\operatorname{grad}g$, $\operatorname{grad}h$ have to follow the same form. Additionally they can be implemented as in-place functions or as allocating ones. The gradient $\operatorname{grad}F$ has to be the same kind. This difference is indicated by the evaluation keyword.

Constructors

ConstrainedProblem(M::AbstractManifold, F, gradF, G, gradG, H, gradH;
evaluation=AllocatingEvaluation()
)

Where F, G, H describe the cost, inequality and equality constraints as described above and gradF, gradG, gradH are the corresponding gradient functions in one of the 4 formats. If the problem does not have inequality constraints, you can set G and gradG no nothing. If the problem does not have equality constraints, you can set H and gradH no nothing or leave them out.

ConstrainedProblem(M::AbstractManifold, F, gradF;
evaluation=AllocatingEvaluation()
)

A keyword argument variant of the constructor above, where you can leave out either G and gradG or H and gradH but not both.

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Manopt.VectorConstraintType
VectorConstraint <: ConstraintType

A type to indicate that constraints are implemented a vector of functions, e.g. $g_i(p) ∈ \mathbb R, i=1,…,m$.

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Manopt.get_grad_equality_constraintFunction
get_grad_equality_constraint(P, p, j)

evaluate the gradient of the j th equality constraint $(\operatorname{grad} h(p))_j$ or $\operatorname{grad} h_j(x)$.

Note

For the FunctionConstraint variant of the problem, this function still evaluates the full gradient. For the MutatingEvaluation and FunctionConstraint of the problem, this function currently also calls get_equality_constraints, since this is the only way to determine the number of cconstraints. It also allocates a full tangent vector.

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Manopt.get_grad_equality_constraint!Function
get_grad_equality_constraint!(P, X, p, j)

Evaluate the gradient of the jth equality constraint $(\operatorname{grad} h(x))_j$ or $\operatorname{grad} h_j(x)$ in place of $X$

Note

For the FunctionConstraint variant of the problem, this function still evaluates the full gradient. For the MutatingEvaluation of the FunctionConstraint of the problem, this function currently also calls get_inequality_constraints, since this is the only way to determine the number of cconstraints and allocates a full vector of tangent vectors

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Manopt.get_grad_equality_constraintsFunction
get_grad_equality_constraints(P, p)

eevaluate all gradients of the equality constraints $\operatorname{grad} h(x)$ or $\bigl(\operatorname{grad} h_1(x), \operatorname{grad} h_2(x),\ldots, \operatorname{grad}h_n(x)\bigr)$ of the ConstrainedProblem P at p.

Note

for the MutatingEvaluation and FunctionConstraint variant of the problem, this function currently also calls get_equality_constraints, since this is the only way to determine the number of cconstraints.

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Manopt.get_grad_equality_constraints!Function
get_grad_equality_constraints!(P, X, p)

evaluate all gradients of the equality constraints $\operatorname{grad} h(p)$ or $\bigl(\operatorname{grad} h_1(p), \operatorname{grad} h_2(p),\ldots,\operatorname{grad} h_n(p)\bigr)$ of the ConstrainedProblem $P$ at $p$ in place of X, which is a vector ofn tangent vectors.

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Manopt.get_grad_inequality_constraintFunction
get_grad_inequality_constraint(P, p, i)

Evaluate the gradient of the i th inequality constraints $(\operatorname{grad} g(x))_i$ or $\operatorname{grad} g_i(x)$.

Note

For the FunctionConstraint variant of the problem, this function still evaluates the full gradient. For the MutatingEvaluation and FunctionConstraint of the problem, this function currently also calls get_inequality_constraints, since this is the only way to determine the number of cconstraints.

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Manopt.get_grad_inequality_constraint!Function
get_grad_inequality_constraint!(P, X, p, i)

Evaluate the gradient of the ith inequality constraints $(\operatorname{grad} g(x))_i$ or $\operatorname{grad} g_i(x)$ of the ConstrainedProblem P in place of $X$

Note

For the FunctionConstraint variant of the problem, this function still evaluates the full gradient. For the MutatingEvaluation and FunctionConstraint of the problem, this function currently also calls get_inequality_constraints,

since this is the only way to determine the number of cconstraints. evaluate all gradients of the inequality constraints $\operatorname{grad} h(x)$ or $\bigl(g_1(x), g_2(x),\ldots,g_m(x)\bigr)$ of the ConstrainedProblem $p$ at $x$ in place of X, which is a vector ofm tangent vectors .

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Manopt.get_grad_inequality_constraintsFunction
get_grad_inequality_constraints(P, p)

evaluate all gradients of the inequality constraints $\operatorname{grad} g(p)$ or $\bigl(\operatorname{grad} g_1(p), \operatorname{grad} g_2(p),…,\operatorname{grad} g_m(p)\bigr)$ of the ConstrainedProblem $P$ at $p$.

Note

for the MutatingEvaluation and FunctionConstraint variant of the problem, this function currently also calls get_equality_constraints, since this is the only way to determine the number of cconstraints.

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Manopt.get_grad_inequality_constraints!Function
get_grad_inequality_constraints!(P, X, p)

evaluate all gradients of the inequality constraints $\operatorname{grad} g(x)$ or $\bigl(\operatorname{grad} g_1(x), \operatorname{grad} g_2(x),\ldots,\operatorname{grad} g_m(x)\bigr)$ of the ConstrainedProblem P at p in place of X, which is a vector of $m$ tangent vectors.

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## Proximal Map(s) based problem

Manopt.ProximalProblemType
ProximalProblem <: Problem

specify a problem for solvers based on the evaluation of proximal map(s).

Fields

• M - a Riemannian manifold
• cost - a function $F:\mathcal M→ℝ$ to minimize
• proxes - proximal maps $\operatorname{prox}_{λ\varphi}:\mathcal M→\mathcal M$ as functions (λ,x) -> y, i.e. the prox parameter λ also belongs to the signature of the proximal map.
• number_of_proxes - (length(proxes)) number of proximal Maps, e.g. if one of the maps is a combined one such that the proximal Maps functions return more than one entry per function

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Manopt.get_proximal_mapFunction
get_proximal_map(p,λ,x,i)

evaluate the ith proximal map of ProximalProblem p at the point x of p.M with parameter $λ>0$.

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## Hessian based problem

Manopt.HessianProblemType
HessianProblem <: Problem

specify a problem for hessian based algorithms.

Fields

• M : a manifold $\mathcal M$
• cost : a function $F:\mathcal M→ℝ$ to minimize
• gradient : the gradient $\operatorname{grad}F:\mathcal M → \mathcal T\mathcal M$ of the cost function $F$
• hessian : the hessian $\operatorname{Hess}F(x)[⋅]: \mathcal T_{x} \mathcal M → \mathcal T_{x} \mathcal M$ of the cost function $F$
• precon : the symmetric, positive definite preconditioner (approximation of the inverse of the Hessian of $F$)

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Manopt.get_hessianFunction
get_hessian(p::HessianProblem{T}, q, X)
get_hessian!(p::HessianProblem{T}, Y, q, X)

evaluate the Hessian of a HessianProblem p at the point q applied to a tangent vector X, i.e. $\operatorname{Hess}f(q)[X]$.

The evaluation is done in place of Y for the !-variant. The T=AllocatingEvaluation problem might still allocate memory within. When the non-mutating variant is called with a T=MutatingEvaluation memory for the result is allocated.

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Manopt.get_preconditionerFunction
get_preconditioner(p,x,ξ)

evaluate the symmetric, positive definite preconditioner (approximation of the inverse of the Hessian of the cost function F) of a HessianProblem p at the point xapplied to a tangent vector ξ.

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## Primal dual based problem

Manopt.PrimalDualProblemType
PrimalDualProblem {T, mT <: AbstractManifold, nT <: AbstractManifold} <: AbstractPrimalDualProblem

Describes a Problem for the linearized or exact Chambolle-Pock algorithm.[BergmannHerzogSilvaLouzeiroTenbrinckVidalNunez2020][ChambollePock2011]

Fields

All fields with !! can either be mutating or nonmutating functions, which should be set depenting on the parameter T <: AbstractEvaluationType.

• M, N – two manifolds $\mathcal M$, $\mathcal N$
• cost $F + G(Λ(⋅))$ to evaluate interims cost function values
• linearized_forward_operator!! linearized operator for the forward operation in the algorithm $DΛ$
• linearized_adjoint_operator!! The adjoint differential $(DΛ)^* : \mathcal N → T\mathcal M$
• prox_F!! the proximal map belonging to $f$
• prox_G_dual!! the proximal map belonging to $g_n^*$
• Λ!! – (fordward_operator) the forward operator (if given) $Λ: \mathcal M → \mathcal N$

Either $DΛ$ (for the linearized) or $Λ$ are required usually.

Constructor

LinearizedPrimalDualProblem(M, N, cost, prox_F, prox_G_dual, adjoint_linearized_operator;
linearized_forward_operator::Union{Function,Missing}=missing,
Λ::Union{Function,Missing}=missing,
evaluation::AbstractEvaluationType=AllocatingEvaluation()
)

The last optional argument can be used to provide the 4 or 5 functions as allocating or mutating (in place computation) ones. Note that the first argument is always the manifold under consideration, the mutated one is the second.

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Manopt.PrimalDualSemismoothNewtonProblemType
PrimalDualSemismoothNewtonProblem {T,mT <: AbstractManifold, nT <: AbstractManifold} <: AbstractPrimalDualProblem{T}

Describes a Problem for the Primal-dual Riemannian semismooth Newton algorithm. [DiepeveenLellmann2021]

Fields

• M, N – two manifolds $\mathcal M$, $\mathcal N$
• cost $F + G(Λ(⋅))$ to evaluate interims cost function values
• linearized_operator the linearization $DΛ(⋅)[⋅]$ of the operator $Λ(⋅)$.
• linearized_adjoint_operator The adjoint differential $(DΛ)^* \colon \mathcal N \to T\mathcal M$
• prox_F the proximal map belonging to $f$
• diff_prox_F the (Clarke Generalized) differential of the proximal maps of $F$
• prox_G_dual the proximal map belonging to $g_n^*$
• diff_prox_dual_G the (Clarke Generalized) differential of the proximal maps of $G^\ast_n$
• Λ – the exact forward operator. This operator is required if Λ(m)=n does not hold.

Constructor

PrimalDualSemismoothNewtonProblem(M, N, cost, prox_F, prox_G_dual, forward_operator, adjoint_linearized_operator,Λ)
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Manopt.get_primal_proxFunction
y = get_primal_prox(p::AbstractPrimalDualProblem, σ, x)
get_primal_prox!(p::AbstractPrimalDualProblem, y, σ, x)

Evaluate the proximal map of $F$ stored within AbstractPrimalDualProblem

$$$\operatorname{prox}_{σF}(x)$$$

which can also be computed in place of y.

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Manopt.get_dual_proxFunction
y = get_dual_prox(p::AbstractPrimalDualProblem, n, τ, ξ)
get_dual_prox!(p::AbstractPrimalDualProblem, y, n, τ, ξ)

Evaluate the proximal map of $G_n^*$ stored within AbstractPrimalDualProblem

$$$\operatorname{prox}_{τG_n^*}(ξ)$$$

which can also be computed in place of y.

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Manopt.adjoint_linearized_operatorFunction
X = adjoint_linearized_operator(p::AbstractPrimalDualProblem, m, n, Y)
adjoint_linearized_operator(p::AbstractPrimalDualProblem, X, m, n, Y)

Evaluate the adjoint of the linearized forward operator of $(DΛ(m))^*[Y]$ stored within the AbstractPrimalDualProblem (in place of X). Since $Y∈T_n\mathcal N$, both $m$ and $n=Λ(m)$ are necessary arguments, mainly because the forward operator $Λ$ might be missing in p.

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Manopt.get_differential_dual_proxFunction
y = get_differential_dual_prox(p::PrimalDualSemismoothNewtonProblem, n, τ, ξ, Ξ)
get_differential_dual_prox!(p::PrimalDualSemismoothNewtonProblem, y, n, τ, ξ, Ξ)

Evaluate the differential proximal map of $G_n^*$ stored within PrimalDualSemismoothNewtonProblem

$$$D\operatorname{prox}_{τG_n^*}(ξ)[Ξ]$$$

which can also be computed in place of y`.

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