Proximal gradient method: Difference between revisions

Revision as of 18:04, 9 December 2018

Revision as of 19:00, 26 January 2019

Line 10:

in that the functions <math>f_1, . . . , f_n</math> are used individually so as to yield an easily [[wikt:implementable|implementable]] algorithm.

They are called [[proximal]] because each non [[smooth function]] among <math>f_1, . . . , f_n</math> is involved via its proximity

operator. Iterative Shrinkage thresholding algorithm,<ref>

{{cite news | last1="Daubechies | first1=I | last2=Defrise | first2 = M | last3 = De Mol| first3 = C| title=An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

|journal=A Journal Issued by the Courant Institute of Mathematical Sciences|volume=57 |year=2004|pages=1413-14571413–1457}}</ref>, [[Landweber iteration|projected Landweber]], projected

gradient, [[alternating projection]]s, [[Alternating direction method of multipliers#Alternating direction method of multipliers|alternating-direction method of multipliers]], alternating

split [[Bregman method|Bregman]] are special instances of proximal algorithms. Details of proximal methods are discussed in Combettes and Pesquet.<ref>

Line 72:

\operatorname{argmin}\limits_y \bigg( f(y) + \frac{1}{2} \left\| x-y \right\|_2^2 \bigg)

</math>

and is denoted <math>\operatorname{prox}_f(x)</math>.

: <math>

Line 114:

*Fast Iterative Shrinkage Thresholding Algorithm (FISTA)<ref>

{{cite news | last1="Beck | first1=A | last2=Teboulle | first2 = M | title=A fast iterative shrinkage-thresholding algorithm for linear inverse problems

|journal=SIAM J.Journal on Imaging ScienceSciences|volume=2 |year=2009|pages=183–202}}</ref>

== See also ==

Line 151:

* [http://www.stanford.edu/class/ee364a/ EE364a: Convex Optimization I] and [http://www.stanford.edu/class/ee364b/ EE364b: Convex Optimization II], Stanford course homepages

* [http://www.eecs.berkeley.edu/~elghaoui/Teaching/EE227A/lecture18.pdf EE227A: Lieven Vandenberghe Notes] Lecture 18

* [https://github.com/kul-forbes/ProximalOperators.jl ProximalOperators.jl]: a [[Julia_Julia (programming_languageprogramming language)|Julia]] package implementing proximal operators.

* [https://github.com/kul-forbes/ProximalAlgorithms.jl ProximalAlgorithms.jl]: a [[Julia_Julia (programming_languageprogramming language)|Julia]] package implementing algorithms based on the proximal operator, including the proximal gradient method.

* [http://proximity-operator.net/ Proximity Operator repository]: a collection of proximity operators implemented in [[Matlab]] and [[Python (programming language)|Python]].