formulasearchengine - User contributions [en]

Generalized entropy index

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128.83.226.170: /* Formula */

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Augmented Lagrangian method

2014-12-01T23:13:05Z

128.83.68.90: Found a typo.

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Electric dipole moment

2014-02-14T17:17:35Z

128.83.230.13:

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Normal variance-mean mixture

2014-02-06T19:52:17Z

128.83.226.170:

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Row-major order

2014-01-31T21:39:33Z

128.83.226.170: Added SAS to list of languages using row-major order.

'''Interior point methods''' (also referred to as '''barrier methods''') are a certain class of [[algorithm]]s to solve linear and nonlinear [[convex optimization]] problems.
[[File:karmarkar.png|thumb|200px|right|Example solution]]
The interior point method was invented by [[John von Neumann]].<ref>{{Cite book|first1=George B.|last1= Dantzig |first2= Mukund N. |last2=Thapa|year= 2003|title= Linear Programming 2: Theory and Extensions|publisher=Springer-Verlag}}</ref> Von Neumann suggested a new method of linear programming, using the homogeneous linear system of Gordan (1873) which was later popularized by [[Karmarkar's algorithm]], developed by [[Narendra Karmarkar]] in 1984 for [[linear programming]]. The method consists of a [[self-concordant]] [[barrier function]] used to encode the [[convex set]]. Contrary to the [[Simplex algorithm|simplex method]], it reaches an optimal solution by traversing the interior of the [[feasible region]].

Any convex optimization problem can be transformed into minimizing (or maximizing) a [[linear function]] over a convex set by converting to the [[epigraph form]].<ref>{{cite book|last=Boyd|first=Stephen|last2=Vandenberghe|first2=Lieven|title=Convex Optimization|publisher=[[Cambridge University Press]]|location=Cambridge |year=2004|pages=143|isbn=0-521-83378-7|mr=2061575}}</ref> The idea of encoding the [[candidate solution|feasible set]] using a barrier and designing barrier methods was studied in the early 1960s by, amongst others, Anthony V. Fiacco and Garth P. McCormick. These ideas were mainly developed for general [[nonlinear programming]], but they were later abandoned due to the presence of more competitive methods for this class of problems (e.g. [[sequential quadratic programming]]).

[[Yurii Nesterov]] and [[Arkadi Nemirovski]] came up with a special class of such barriers that can be used to encode any convex set. They guarantee that the number of [[iteration]]s of the algorithm is bounded by a polynomial in the dimension and accuracy of the solution.<ref>{{Cite journal|mr=2115066|doi=10.1090/S0273-0979-04-01040-7|title=The interior-point revolution in optimization: History, recent developments, and lasting consequences|year=2004|last1=Wright|first1=Margaret H.|journal=Bulletin of the American Mathematical Society|volume=42|pages=39 }}</ref>

Karmarkar's breakthrough revitalized the study of interior point methods and barrier problems, showing that it was possible to create an algorithm for linear programming characterized by [[polynomial time|polynomial complexity]] and, moreover, that was competitive with the simplex method.
Already [[Leonid Khachiyan|Khachiyan]]'s [[ellipsoid method]] was a polynomial time algorithm; however, in practice it was too slow to be of practical interest.

The class of primal-dual path-following interior point methods is considered the most successful.
[[Mehrotra predictor-corrector method|Mehrotra's predictor-corrector algorithm]] provides the basis for most implementations of this class of methods{{Citation needed|date=February 2011}}.

==Primal-dual interior point method for nonlinear optimization==
The primal-dual method's idea is easy to demonstrate for constrained [[nonlinear optimization]].
For simplicity consider the all-inequality version of a nonlinear optimization problem:

:minimize <math>f(x)</math> subject to <math>c(x) \ge 0~~ x \in \mathbb{R}^n, c(x) \in \mathbb{R}^m~~~~~~(1)</math>.

The logarithmic [[barrier function]] associated with (1) is
:<math>B(x,\mu) = f(x) - \mu~ \sum_{i=1}^m\ln(c_i(x))~~~~~(2)</math>

Here <math>\mu</math> is a small positive scalar, sometimes called the "barrier parameter". As <math>\mu</math> converges to zero the minimum of <math>B(x,\mu)</math> should converge to a solution of (1).

The barrier function [[gradient]] is
:<math>g_b = g - \mu\sum_{i=1}^m \frac{1}{c_i(x)} \nabla c_i(x)~~~~~~(3)</math>

where <math>g</math> is the gradient of the original function <math>f(x)</math> and <math>\nabla c_i</math> is the gradient of <math>c_i</math>.

In addition to the original ("primal") variable <math>x</math> we introduce a [[Lagrange multiplier]] inspired [[Lagrange_multiplier#The_strong_Lagrangian_principle:_Lagrange_duality|dual]] variable <math>\lambda\in \mathbb{R} ^m</math>(sometimes called "slack variable")
:<math>\forall_{i=1}^m c_i(x) \lambda_i=\mu~~~~~~~(4)</math>

(4) is sometimes called the "perturbed complementarity" condition, for its resemblance to "complementary slackness" in [[KKT conditions]].

We try to find those <math>(x_\mu, \lambda_\mu)</math> for which the gradient of the barrier function is zero.

Applying (4) to (3) we get equation for gradient:
:<math>g - A^T \lambda = 0~~~~~~(5)</math>
where the matrix <math>A</math> is the constraint <math>c(x)</math> [[Jacobian matrix and determinant|Jacobian]].

The intuition behind (5) is that the gradient of <math>f(x)</math> should lie in the subspace spanned by the constraints' gradients. The "perturbed complementarity" with small <math>\mu</math> (4) can be understood as the condition that the solution should either lie near the boundary <math>c_i(x) = 0</math> or that the projection of the gradient <math>g</math> on the constraint component <math>c_i(x)</math> normal should be almost zero.

Applying [[Newton method|Newton's method]] to (4) and (5) we get an equation for <math>(x, \lambda)</math> update <math>(p_x, p_\lambda)</math>:
:<math>\begin{pmatrix}
W & -A^T \\
\Lambda A & C
\end{pmatrix}\begin{pmatrix}
p_x \\
p_\lambda
\end{pmatrix}=\begin{pmatrix}
-g + A^T \lambda \\
\mu 1 - C \lambda
\end{pmatrix}</math>

where <math>W</math> is the [[Hessian matrix]] of <math>f(x)</math> and <math>\Lambda</math> is a [[diagonal matrix]] of <math>\lambda</math>.

Because of (1), (4) the condition
:<math>\lambda \ge 0</math>

should be enforced at each step. This can be done by choosing appropriate <math>\alpha</math>:
:<math>(x,\lambda) \rightarrow (x+ \alpha p_x, \lambda + \alpha p_\lambda)</math>.

==See also==
*[[Augmented Lagrangian method]]
*[[Penalty method]]
*[[Karush–Kuhn–Tucker conditions]]

==References==
{{Reflist}}
== Bibliography ==
* {{cite book|last1=Bonnans|first1=J. Frédéric|last2=Gilbert|first2=J. Charles|last3=Lemaréchal|first3=Claude| authorlink3=Claude Lemaréchal|last4=Sagastizábal|first4=Claudia A.|title=Numerical optimization: Theoretical and practical aspects|url=http://www.springer.com/mathematics/applications/book/978-3-540-35445-1|edition=Second revised ed. of translation of 1997  French| series=Universitext|publisher=Springer-Verlag|location=Berlin|year=2006|pages=xiv+490|isbn=3-540-35445-X|doi=10.1007/978-3-540-35447-5|mr=2265882}}
* {{cite journal|doi=10.1145/800057.808695|url=http://retis.sssup.it/~bini/teaching/optim2010/karmarkar.pdf|chapter=A new polynomial-time algorithm for linear programming|title=Proceedings of the sixteenth annual ACM symposium on Theory of computing - STOC '84|year=1984|last1=Karmarkar|first1=N.|isbn=0-89791-133-4|pages=302}}
* {{Cite journal|doi=10.1137/0802028|title=On the Implementation of a Primal-Dual Interior Point Method|year=1992|last1=Mehrotra|first1=Sanjay|journal=SIAM Journal on Optimization|volume=2|issue=4|pages=575}}
*{{cite book|title = Numerical Optimization | first=Jorge| last = Nocedal | coauthors= and Stephen Wright| year=1999 | publisher=Springer | location=New York, NY| isbn=0-387-98793-2}}
*{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 10.11. Linear Programming: Interior-Point Methods | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=537}}
*{{cite book|title = Primal-Dual Interior-Point Methods | first=Stephen| last = Wright | year=1997 | publisher=SIAM | location=Philadelphia, PA| isbn=0-89871-382-X}}
*{{cite book|title = Convex Optimization |last1=Boyd|first1=Stephen|last2=Vandenberghe|first2=Lieven|year=2004|publisher=Cambridge University Press|url=http://www.stanford.edu/~boyd/cvxbook/}}

{{Use dmy dates|date=February 2011}}

{{Optimization algorithms|convex}}

[[Category:Optimization algorithms and methods]]

Radius of curvature (applications)

2014-01-07T20:40:15Z

128.83.167.210: /* Radius of curvature applied to measurements of the stress in the semiconductor structures */

In [[computing]], a '''Las Vegas algorithm''' is a [[randomized algorithm]] that always gives [[correctness (computer science)|correct]] results; that is, it always produces the correct result or it informs about the failure. In other words, a Las Vegas algorithm does not gamble with the correctness of the result; it gambles only with the resources used for the computation. A simple example is randomized [[quicksort]], where the pivot is chosen randomly, but the result is always sorted. The usual definition of a Las Vegas algorithm includes the restriction that the ''expected'' run time always be finite, when the expectation is carried out over the space of random information, or entropy, used in the algorithm.

Las Vegas algorithms were introduced by [[László Babai]] in 1979, in the context of the [[graph isomorphism problem]], as a stronger version of [[Monte Carlo algorithm]]s.<ref>[[László Babai]], [http://people.cs.uchicago.edu/~laci/lasvegas79.pdf Monte-Carlo algorithms in graph isomorphism testing], Université de Montréal, D.M.S. No. 79-10.</ref><ref>[[Leonid Levin]], [http://arxiv.org/abs/cs.CR/0012023 The Tale of One-way Functions], ''Problems of Information Transmission'', vol. 39 (2003), 92-103.</ref><ref>Dan Grundy, [http://www.cs.kent.ac.uk/people/staff/eab2/crypto/thesis.web.pdf Concepts and Calculation in Cryptography], University of Kent, Ph.D. thesis, 2008</ref> Las Vegas algorithms can be used in situations where the number of possible solutions is relatively limited, and where verifying the correctness of a candidate solution is relatively easy while actually calculating the solution is complex.

The name refers to the city of [[Las Vegas, Nevada]], which is well known within the United States as an icon of gambling.

== Complexity class ==

The [[complexity class]] of [[decision problem]]s that have Las Vegas algorithms with [[expected value|expected]] polynomial runtime is '''[[Zero-error_Probabilistic_Polynomial_time | ZPP]]'''.

It turns out that

:<math>\textrm{ZPP} = \textrm{RP} \cap \,\text{co}\,\textrm{-RP}, \,\!</math>

which is intimately connected with the way Las Vegas algorithms are sometimes constructed. Namely the class '''[[RP (complexity)|RP]]''' consists of all decision problems for which a randomized polynomial-time algorithm exists that always answers correctly when the correct answer is "no", but is allowed to be wrong with a certain probability bounded away from one when the answer is "yes". When such an algorithm exists for both a problem and its complement (with the answers "yes" and "no" swapped), the two algorithms can be run simultaneously and repeatedly: run each for a constant number of steps, taking turns, until one of them returns a definitive answer. This is the standard way to construct a Las Vegas algorithm that runs in expected polynomial time. Note that in general there is no worst case upper bound on the run time of a Las Vegas algorithm.

== Relation to Monte Carlo algorithms ==

Las Vegas algorithms can be contrasted with [[Monte Carlo algorithm]]s, in which the resources used are bounded but the answer is not guaranteed to be correct 100% of the time. By an application of [[Markov's inequality]], a Las Vegas algorithm can be converted into a Monte Carlo algorithm via early termination.

== See also ==
* [[Randomness]]
* [[Monte Carlo algorithm]]

==Notes ==
{{reflist}}

==References==
* ''Algorithms and Theory of Computation Handbook'', CRC Press LLC, 1999, "Las Vegas algorithm", in ''Dictionary of Algorithms and Data Structures'' [online], Paul E. Black, ed., U.S. [[National Institute of Standards and Technology]]. 17 July 2006. (accessed May 09, 2009) Available from: [http://www.nist.gov/dads/HTML/lasVegas.html]

[[Category:Randomized algorithms]]

Fama–French three-factor model

2012-07-26T14:49:42Z

128.83.184.60:

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Woodbury matrix identity

2012-06-23T00:04:13Z

128.83.68.153: /* See also */

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Trigonometric moment problem

2011-04-14T21:41:22Z

128.83.207.76: /* Parametrization of solutions */

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