en>Funandtrvl: nvbx

2013-06-28T23:50:40Z

nvbx

← Older revision		Revision as of 01:50, 29 June 2013
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	~~Alyson~~ is ~~what my husband enjoys~~ to ~~contact me but I don~~'~~t like when individuals use my full title~~. ~~One of~~ the ~~issues she loves most~~ is ~~canoeing~~ and ~~she's been performing it for fairly~~ a ~~while~~. ~~I am~~ an ~~invoicing officer~~ and I'll be ~~promoted soon~~. ~~North Carolina~~ is ~~exactly where we've been living for many years~~ and ~~will by no means move~~.<br><br>~~My website~~ :~~: real~~ [http://~~165~~.~~132~~.39.~~93/xe/visitors/372912 online psychic readings~~] [http://si.~~dgmensa~~.~~org~~/xe/~~index~~.~~php?document_srl~~=~~48014&mid~~=~~c0102 phone psychic readings] (~~[http://~~alles~~-~~herunterladen~~.de/~~excellent~~-~~advice~~-~~for~~-~~picking-the-ideal~~-~~hobby/ check out here])~~		In [[mathematical optimization]], a '''quadratically constrained quadratic program''' ('''QCQP''') is an [[optimization problem]] in which both the objective function and the constraints are quadratic functions. It has the form

			:<math> \begin{align}
			& \text{minimize} && \tfrac12 x^\mathrm{T} P_0 x + q_0^\mathrm{T} x \\
			& \text{subject to} && \tfrac12 x^\mathrm{T} P_i x + q_i^\mathrm{T} x + r_i \leq 0 \quad \text{for } i = 1,\dots,m , \\
			&&& Ax = b,
			\end{align} </math>

			where ''P''<sub>0</sub>, … ''P''<sub>''m''</sub> are ''n''-by-''n'' matrices and ''x'' ∈ '''R'''<sup>''n''</sup> is the optimization variable.

			If ''P''<sub>0</sub>, … ''P''<sub>''m''</sub> are all [[Positive-definite matrix\|positive semidefinite]], then the problem is [[Convex set\|convex]]. If these matrices are neither positive or negative semidefinite, the problem is non-convex. If ''P''<sub>1</sub>, … ''P''<sub>''m''</sub> are all zero, then the constraints are in fact [[Linear map\|linear]] and the problem is a [[quadratic program]].

			== Hardness ==
			Solving the general case is an [[NP-hard]] problem. To see this, note that the two constraints ''x''<sub>1</sub>(''x''<sub>1</sub> − 1) ≤ 0 and ''x''<sub>1</sub>(''x''<sub>1</sub> − 1) ≥ 0 are equivalent to the constraint ''x''<sub>1</sub>(''x''<sub>1</sub> − 1) = 0, which is in turn equivalent to the constraint ''x''<sub>1</sub> ∈ {0, 1}. Hence, any [[0–1 integer program]] (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained quadratic program. Since 0–1 integer programming is NP-hard in general, QCQP is also NP-hard.

			== Relaxation ==
			There are two main relaxations of QCQP: using [[semidefinite programming]] (SDP), and using the [[reformulation-linearization technique]] (RLT).

			=== Semidefinite programming ===
			When ''P''<sub>0</sub>, … ''P''<sub>''m''</sub> are all [[Positive-definite matrix\|positive-definite matrices]], the problem is [[Convex optimization\|convex]] and can be readily solved using [[interior point method]]s, as done with [[semidefinite programming]].

			== Example ==
			[[Cut (graph theory)\|Max Cut]] is a problem in graph theory, which is NP-hard. Given a graph, the problem is to divide the vertices in two sets, so that as many edges as possible go from one set to the other. Max Cut can be formulated as a QCQP, and SDP relaxation of the dual provides good lower bounds.

			==Solvers and scripting (programming) languages==

			{\| class="wikitable"
			\|-
			!Name
			!Brief info
			\|-
			\|[[AMPL]]\|\|
			\|-
			\|[[CPLEX]]\|\| Popular solver with an API for several programming languages. Free for academics.
			\|-
			\|[[Gurobi]]\|\| Solver with parallel algorithms for large-scale linear programs, quadratic programs and mixed-integer programs. Free for academic use.
			\|-
			\|JOptimizer\|\| Java library for convex optimization (open source)
			\|-
			\|[[MOSEK]]\|\| A solver for large scale optimization with API for several languages (C++,java,.net, Matlab and python)
			\|-
			\|[[OpenOpt]]\|\|universal cross-platform numerical optimization framework, see its [http://openopt.org/QCQP QCQP] page and [http://openopt.org/Problems other problems] involved. Uses [[NumPy]] arrays and [[SciPy]] sparse matrices.
			\|-
			\|[[TOMLAB]]\|\|Supports global optimization, integer programming, all types of least squares, linear, quadratic and unconstrained programming for [[MATLAB]]. TOMLAB supports solvers like [[Gurobi]], [[CPLEX]], [[SNOPT]] and [[KNITRO]].
			\|}

			== References ==
			* {{cite book
			\| last = Boyd
			\| first = Stephen
			\| coauthors = Lieven Vandenberghe
			\| title = Convex Optimization
			\| publisher = Cambridge University Press
			\| year = 2004
			\| location = Cambridge
			\| url = http://www.stanford.edu/~boyd/cvxbook/
			\| isbn = 0-521-83378-7}}

			== Further reading ==

			=== In statistics ===
			* {{cite journal \|author=Albers CJ, Critchley F, Gower, JC \|title=Quadratic Minimisation Problems in Statistics \|journal=Journal of Multivariate Analysis \|volume=102 \|issue=3 \|pages=698–713 \|year=2011 \|doi=10.1016/j.jmva.2009.12.018 }}

			==External links==
			* [http://neos-guide.org/content/quadratic-constrained-quadratic-programming NEOS Optimization Guide: Quadratic Constrained Quadratic Programming]

			[[Category:Mathematical optimization]]


			{{mathapplied-stub}}

en>Boriaj: /* Main properties */ corrected TeX error.

2012-06-02T19:52:12Z

Main properties: corrected TeX error.

New page

Alyson is what my husband enjoys to contact me but I don't like when individuals use my full title. One of the issues she loves most is canoeing and she's been performing it for fairly a while. I am an invoicing officer and I'll be promoted soon. North Carolina is exactly where we've been living for many years and will by no means move.<br><br>My website :: real [http://165.132.39.93/xe/visitors/372912 online psychic readings] [http://si.dgmensa.org/xe/index.php?document_srl=48014&mid=c0102 phone psychic readings] ([http://alles-herunterladen.de/excellent-advice-for-picking-the-ideal-hobby/ check out here])

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en>Funandtrvl: nvbx

en>Boriaj: /* Main properties */ corrected TeX error.