ConstrainedDFO.jl

AbstractInvertibilityBound

A formula to compute a lower bound on the invertibility_radius of a manifold.

AbstractTangentSolver

An abstract type for all solvers used to solve subproblems in tangent spaces with the DFROSolver.

BlackboxInstance

BlackboxProblem

BlackboxTangentData

A structure representing all the data expected to be returned by an AbstractTangentSolver evaluating a blackbox.

DFROState <: AbstractManoptSolverState

AbstractManoptSolverState dedicated to the RDFO solver.

Fields

• p is the current iterate on the manifold.
• d is the current best tangent vector found at $T_p\mathcal{M}$.

EqualityManifold <: AbstractManifold{ℝ}

A smooth Riemannian submanifold of $\mathbb{R}^n$ defined as the set

\[ \mathcal{M}=\left\{x\in\mathbb{R}^n : h(x)=0\right\}\]

for some smooth function $h: \mathbb{R}^n\to\mathbb{R}$ such that $\nabla h(x)$ has full rank for all $x\in\mathcal{M}$.

Fields

• defining_function: the function $h$ as described above.
• dimension: the dimension of the manifold, defined as the common dimension of its tangent spaces.

ExactInvertibility <: AbstractInvertibilityBound

Computes the exact value of the invertibility_radius of a retraction (if possible).

MADSTangentSolver <: AbstractTangentSolver

Subsolver using the Mesh Adaptive Direct Search (MADS) algorithm in tangent spaces. To be used together with the master solver DFROSolver.

Note: the implementation makes use of the interface to the NOMAD 3 software offered by NOMAD.jl

NOverSpectral <: AbstractInvertibilityBound

Computes a lower bound on the invertibility_radius of the ProjectionRetraction as

\[ \frac{n}{\max\{\lambda(H_{h_i}(x)) : i\in\{1,...,p\}\}}.\]

with $\lambda(H_{h_i})$ the spectral radius of the Hessian matrix of the defining subfunction $h_i$ for M.

NOverSqrtSpectral <: AbstractInvertibilityBound

Computes a lower bound on the invertibility_radius of the ProjectionRetraction as

\[ \frac{n}{\sqrt{\max\{\lambda(H_{h_i}(x)) : i\in\{1,...,p\}\}}}.\]

with $\lambda(H_{h_i})$ the spectral radius of the Hessian matrix of the defining subfunction $h_i$ for M.

OneOverSpectral <: AbstractInvertibilityBound

Computes a lower bound on the invertibility_radius of the ProjectionRetraction as

\[ \frac{1}{\max\{\lambda(H_{h_i}(x)) : i\in\{1,...,p\}\}}.\]

with $\lambda(H_{h_i})$ the spectral radius of the Hessian matrix of the defining subfunction $h_i$ for M.

OneOverSqrtSpectral <: AbstractInvertibilityBound

Computes a lower bound on the invertibility_radius of the ProjectionRetraction as

\[ \frac{1}{\sqrt{\max\{\lambda(H_{h_i}(x)) : i\in\{1,...,p\}\}}}.\]

with $\lambda(H_{h_i})$ the spectral radius of the Hessian matrix of the defining subfunction $h_i$ for M.

StopAfterEvaluation <: DFStoppingCriterion

A functor for a stopping criterion to stop after a maximal number of blackbox evaluations. Fields and constructor are the same as StopAfterIteration.

DFROSolver(
    M::AbstractManifold,
    f::Function,
    p0;
    inequality_constraints::Union{Function, Nothing} = nothing,
    solver::AbstractTangentSolver = MADSTangentSolver(),
    max_evals::Int = 1000 * representation_size(M)[1],
    stopping_criterion::DFStoppingCriterion = StopRadiusAndBudget(max_evals),
    retraction_method::AbstractRetractionMethod = default_retraction_method(M),
    invertibility_bound::AbstractInvertibilityBound = default_invertibility_bound(M, retraction_method),
    εeqs::Float64 = 1.0e-8
)

blackbox_wrapper_store!(TS::AbstractTangentSolver, M::AbstractManifold, p, R::AbstractRetractionMethod, f, n_ineqs::Int, g, v)

Retract the tangent vector v to the manifold M and evaluate the blackbox made of the objective function f and the inequality constraints g. The results are stored within corresponding attributes of the AbstractTangentSolver.

default_invertibility_bound(M::AbstractManifold; m::AbstractRetractionMethod)

Return the default AbstractInvertibilityBound used to compute the invertibility_radius of $M$ when endowed with the retraction method m. For an EqualityManifold endowed with the ProjectionRetraction, defaults to NOverSpectral.

get_eval_budget(em::AbstractEvalManager)

get_last_subproblem_result(TS::AbstractTangentSolver)

Retrieve all data stored within the AbstractTangentSolver object. Contains:

• values of all evaluated points within $\mathbb{R}^q$;
• associated retractions on the manifold the solver was last called on;
• associated values of the objective function and inequality constraints.

invertibility_radius(M::AbstractManifold, p; m::AbstractRetractionMethod, ρ::AbstractInvertibilityBound)

When the manifold $M$ is endowed with retraction method m at p, its invertibility radius is defined as

\[ \mathrm{inv}(p)=\sup\{\delta>0\; :\; R_p\text{ is a diffeomorphism from }B_p(0;\delta)\text{ onto its image}\}.\]

This function returns a lower bound on this quantity, computed according to ρ. If ρ is an ExactInvertibility, the exact value is returned.

When m is the ExponentialRetraction, this function falls back to the injectivity radius of $M$ at p.

latin_hypercube_sampling(n, lb, ub, p)

Perform a latin hypercube sampling with given bounds, and return p sampled points. It is assumed that bounds are well-defined.

solve!(TS::AbstractTangentSolver, f, M::AbstractManifold, p, R::AbstractRetractionMethod, ρ::AbstractInvertibilityBound; g)

Solve the subproblem

\[ \begin{array}{r l l} \min\limits_{v\in T_p\mathcal{M}} & f\circ R_p(v) \\ \mathrm{s.t.} & g\circ R_p(v) \leq 0 \end{array}\]

with the tangent solver TS. Stops whenever an iterate (i.e., a feasible improving point) is found outside of the invertibility_radius, with bound given by ρ (see AbstractInvertibilityBound).

A history of all tangent iterates, associated retractions and (f,g) values is stored within the AbstractTangentSolver object.

update_remaining_evals(dem::DefaultEvalManager)