## PartitionedStructures.jl

Partitioned derivatives storage and partitioned quasi-Newton updates
Author JuliaSmoothOptimizers
Popularity
9 Stars
Updated Last
1 Year Ago
Started In
March 2021

# PartitionedStructures.jl: Partitioned derivatives storage and partitioned quasi-Newton updates

Documentation Linux/macOS/Windows/FreeBSD Coverage DOI

## How to cite

If you use PartitionedStructures.jl in your work, please cite using the format given in CITATION.bib.

Julia ≥ 1.6.

## How to install

pkg> add PartitionedStructures
pkg> test PartitionedStructures

## Philosophy

Methods exploiting the derivatives of partially-separable functions require specific data structures to store partitioned derivatives. There are several types of partial separability. We write a partially-separable function $f: \mathbb{R}^n \to \mathbb{R}$ in the form

$$f(x) = \sum_{i=1}^N f_i (U_i(x)), f_i : \mathbb{R}^{n_i} \to \mathbb{R}, U_i \in \mathbb{R}^{n_i \times n}, n_i \ll n,$$

where:

• $f_i$ is the $i$-th element function whose dimension is smaller than $f$;
• $U_i$ the linear operator selecting the linear combinations of variables that parametrize $f_i$.

In the case of partitioned quasi-Newton methods, they require storing partitioned gradients and the partitioned Hessian approximation. PartitionedStructures.jl facilitates the definition of those partitioned structures and defines methods to manipulate them.

## Features

$U_i$ may be based on the elemental variables or the internal variables of $f_i$:

• the elemental variables represent the subset of variables that parametrizes $f_i$, i.e. the rows of $U_i$ are vectors from the Euclidean basis;
Ui = [1,3,5] # i.e. [1 0 0 0 0; 0 0 1 0 0; 0 0 0 0 1]

• the internal variables are linear combinations of the variables that parametrize $f_i$, i.e. $U_i$ may be a dense matrix.

The implementation of the linear-operator $U_i$, which describe entirely the partially-separable structure of $f$, changes depending on wether we use internal or elemental variables.

At the moment, we only developed the elemental partitioned structures, but we left the door open to the development of internal partitioned structures in the future.

## How to use

Check the tutorial.

## Partitioned structures available

Structure Description
AbstractPartitionedStructure The supertype of every partitioned structures
Elemental_pm An elemental partitioned matrix, each element-matrix is dense
Elemental_plo_bfgs A limited-memory elemental partitioned matrix, each element limited-memory operator is a LBFGSOperator
Elemental_plo_sr1 A limited-memory elemental partitioned matrix, each element limited-memory operator is a LSR1Operator
Elemental_plo A limited-memory elemental partitioned matrix, each element limited-memory operator is a LBFGSOperator or a LSR1Operator
Elemental_pv An elemental partitioned vector

## Methods available

Method Description
identity_epm Create a partitioned matrix with identity element-matrices
identity_eplo_LBFGS Create a PLBFGS limited-memory partitioned matrix
identity_eplo_LSR1 Create a PLSR1 limited-memory partitioned matrix
identity_eplo_LOSE Create a PLSE limited-memory partitioned matrix
update Performs a partitioned quasi-Newton update on a partitioned matrix
eplo_lbfgs_from_epv Create an Elemental_plo_bfgs from the partitioned structure of an Elemental_pv
eplo_lsr1_from_epv Create an Elemental_plo_sr1 from the partitioned structure of an Elemental_pv
eplo_lose_from_epv Create an Elemental_plo from the partitioned structure of an Elemental_pv
epm_from_epv Create an Elemental_pm from the partitioned structure of an Elemental_pv
epv_from_epm Create an Elemental_pv from the partitioned structure of an Elemental_pm
epv_from_eplo Create an Elemental_pv from the partitioned structure of an Elemental_plo, an Elemental_plo_bfgs or an Elemental_plo_sr1
mul_epm_epv Return a partitioned vector from an elementwise product between a partitioned matrix and a partitioned vector
mul_epm_vector Return the vector resulting from a partitioned matrix-vector product
build_v! Build a vector accumulating the element contributions of a partitioned vector
set_epv! Set the value of every element-vector
minus_epv! Apply a unary minus on every element-vector of a partitioned vector
add_epv! Perform elementwise addition between two partitioned vectors

## Modules using PartitionedStructures.jl

The structures defined here are used in the modules:

# Bug reports and discussions

If you think you found a bug, feel free to open an issue. Focused suggestions and requests can also be opened as issues. Before opening a pull request, start an issue or a discussion on the topic, please.

If you want to ask a question not suited for a bug report, feel free to start a discussion here. This forum is for general discussion about this repository and the JuliaSmoothOptimizers, so questions about any of our packages are welcome.