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{{distinguish|Dijkstra's algorithm}}
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'''Dykstra's algorithm''' is a method that computes a point in the intersection of [[convex set]]s, and is a variant of the [[alternating projection]] method (also called the [[projections onto convex sets]] method). In its simplest form, the method finds a point in the intersection of two convex sets by iteratively projecting onto each of the convex set; it differs from the alternating projection method in that there are intermediate steps. A parallel version of the algorithm was developed by Gaffke and Mathar.
 
The method is named after R. L. Dykstra who proposed it in the 1980s.
 
A key difference between Dykstra's algorithm and the standard alternating projection method occurs when there is more than one point in the intersection of the two sets. In this case, the alternating projection method gives some arbitrary point in this intersection, whereas Dykstra's algorithm gives a specific point: the projection of ''r'' onto the intersection, where ''r'' is the initial point used in the algorithm,
 
== Algorithm ==
Dykstra's algorithm solves the following problem:
 
: <math> \text{minimize}_{x \in \mathcal{R}^n} \|x-r\|^2 \quad\text{subject to}\; x \in C \cap D </math>
 
where <math>C,D</math> are [[convex set]]s.  This problem is equivalent to finding the [[Projection (mathematics)|projection]] of <math>r</math> onto the set <math>C\cap D</math>, which we denote by <math>\mathcal{P}_{C \cap D}</math>.
 
To use Dykstra's algorithm, one must know how to project onto the sets <math>C</math> and <math>D</math> separately.
 
First, consider the basic [[alternating projection]] (aka POCS) method (first studied, in the case when the sets <math>C,D</math> were linear subspaces, by [[John von Neumann]]<ref>J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50 (1949) 401–485 (a reprint of lecture notes first distributed in 1933).</ref>), which initializes <math>x_0=r</math> and then generates the sequence
 
: <math>x_{k+1} = \mathcal{P}_C \left( \mathcal{P}_D ( x_k ) \right) </math>.
 
Dykstra's algorithm is of a similar form, but uses additional auxiliary variables. Start with <math>x_0 =r, p_0=q_0=0</math> and update by
 
: <math> y_k = \mathcal{P}_D( x_k + p_k ) </math>
 
: <math> p_{k+1} = x_k + p_k - y_k </math>
 
: <math> x_{k+1} = \mathcal{P}_C( y_k + q_k ) </math>
 
: <math> q_{k+1} =y_k + q_k - x_{k+1}. </math>
 
Then the sequence <math>(x_k)</math> converges to the solution of the original problem. For convergence results and a modern perspective on the literature, see <ref>P. L. Combettes and J.-C. Pesquet, "Proximal splitting methods in signal processing," in: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, (H. H. Bauschke, R. S. Burachik, P. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, Editors), pp. 185–212. Springer, New York, 2011 [http://www.ann.jussieu.fr/~plc/prox.pdf]</ref>
 
== References ==
 
* {{cite journal | first1 = J. P. | last1 = Boyle | first2 = R. L. | last2 = Dykstra | title = A method for finding projections onto the intersection of convex sets in Hilbert spaces | journal = Lecture Notes in Statistics | volume = 37 | year = 1986 | pages = 28–47}}
* {{cite journal | first1 = N. | last1 = Gaffke | first2 = R. | last2 = Mathar | title = A cyclic projection algorithm via duality | journal = Metrika | volume = 36 | year = 1989 | pages = 29–54}}
<references>
</references>
 
[[Category:Convex geometry]]
[[Category:Optimization algorithms and methods]]

Revision as of 09:53, 1 March 2014

Now, we've organized this brief mention how we can quickly concept a new most effective chef knives that will not alone accomplish but furthermore get past the made intent with the meals prep. Moreover, this is a very good knife for cutting via really hard supplies like bones in chickens. A kitchen knife gets utilised a lot, so make you food preparation simpler by buying excellent ones. For a starter kitchen, however, the paring knife, chef's knife, serrated knife and cleaver are very good standard equipment. You can make extra space in your drawers and hold your new chef knife set on the counter where you can discover it conveniently.

As a suggests of assisting those seeking for the greatest paring knife or ceramic knives, the short article beneath compares the 5 most effective knives, permitting you to make an informed decision. The Kontera chef's knife will let you to chop, dice and slice with ease and won't mar the food in any way. Most will come across that the paring knife will take up the main body of work - paring knives are hugely versatile. His prior knife was a Western Style Ito Sei Santouko.

It is important to obtain a chef's knife that fits the hand nicely. Most superior kitchen stores also have staff on hand to help you pick out the ideal chef's knife for you. A boning knife is four to 5 inches long and has a flexible blade for cutting about bone and meat. If you adored this short article and you would such as to obtain additional details concerning Best Brand Kitchen Knives kindly see our own site. Cut and chop slowly and cautiously when 1st studying to use a knife. Hands, cutting surfaces and the knife must be dry to keep away from slipping. The second knife you will need to have is a excellent paring knife.

The sort of particular person who genuinely gets the very best out of a knife set is somebody who loves to prepare nice meals from fresh components, who has a kitchen that they adore and who definitely basks in how homey and comfortable or sleek and fashionable it is. Ablock performs great for a person who is really busy, and who doesn't mind making dinner, but knows that in order to make it additional enjoyable, they will need to make the prep perform run smoother.

There will be several sorts of knives integrated in every knife set. Identifying the particular knives every single set contains will be incredibly useful in deciding upon the 1 that will very best serve your requires. Chef's knife - Also known as the French knife, it is the most versatile knife in a chef's or culinary student's collection. Paring knife - Mainly applied for trimming and paring fruits and vegetables, it is the smallest knife in the collection with a two- to four-inch blade. Verify out the tako sashimi knife.

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