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A manifold objective

Within the optimization problem

\[\operatorname*{argmin}_{p \in \mathcal M} f(p)\]

the objective describes the cost $f(p)$ and its properties relations. The general abstract type for these is

Manopt.AbstractManifoldObjective — Type
AbstractManifoldObjective{E<:AbstractEvaluationType}

Describe the collection of the optimization function $f: \mathcal{M} → ℝ$ (or even a vectorial range) and its corresponding elements, which might for example be a gradient or (one or more) proximal maps.

All these elements should usually be implemented as functions (M, p) -> ..., or (M, X, p) -> ... that is

  • the first argument of these functions should be the manifold M they are defined on
  • the argument X is present, if the computation is performed in-place of X (see InplaceEvaluation)
  • the argument p is the place the function ($f$ or one of its elements) is evaluated at.

the type T indicates the global AbstractEvaluationType.

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For components of the objective, there might be different ways they can be evaluated, most prominently, whether for example a function providing the gradient $\operatorname{grad} f$ would allocate new memory or work inplace.

Manopt.AllocatingInplaceEvaluation — Type
AllocatingInplaceEvaluation <: AbstractEvaluationType

A parameter for a AbstractManoptProblem or a Function indicating that the problem contains or the function(s) that provides both an allocating variant and one, that does not allocate memory but work on their input, in place.

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The different types of objectives are listed on sub pages depending on their type of information or function