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| {{Context|date=October 2009}}
| | over the counter std test ([http://www.hotporn123.com/user/RPocock official statement]) writer is called Irwin. Puerto Rico is exactly where he's always been living but she requirements to move because of her family. Playing baseball is the pastime he will by no means quit performing. I am a meter reader. |
| {{Unreferenced|date=October 2006}}
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| '''Sequential quadratic programming''' ('''SQP''') is an [[iterative method]] for [[nonlinear programming|nonlinear optimization]]. SQP methods are used on problems for which the [[objective function]] and the constraints are twice [[continuously differentiable]].
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| SQP methods solve a sequence of optimization subproblems, each of which optimizes a quadratic model of the objective subject to a linearization of the constraints. If the problem is unconstrained, then the method reduces to [[Newton's method]] for finding a point where the gradient of the objective vanishes. If the problem has only equality constraints, then the method is equivalent to applying [[Newton's method]] to the first-order optimality conditions, or [[Karush–Kuhn–Tucker conditions]], of the problem. SQP methods have been implemented in many packages, including [[NPSOL]], [[SNOPT]], [[NLPQL]], [[OPSYC]], [[OPTIMA]], [[MATLAB]] and SQP.
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| ==Algorithm basics==
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| Consider a [[nonlinear programming]] problem of the form:
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| :<math>\begin{array}{rl}
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| \min\limits_{x} & f(x) \\
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| \mbox{s.t.} & b(x) \ge 0 \\
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| & c(x) = 0.
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| \end{array}</math>
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| The Lagrangian for this problem is;
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| :<math>\mathcal{L}(x,\lambda,\sigma) = f(x) - \lambda^T b(x) - \sigma^T c(x),</math> | |
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| where <math>\lambda</math> and <math>\sigma</math> are [[Lagrange multipliers]]. At an iterate <math>x_k</math>, a basic sequential quadratic programming algorithm defines an appropriate search direction <math>d_k</math> as a solution to the [[quadratic programming]] subproblem
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| :<math>\begin{array}{rl} \min\limits_{d} & f(x_k) + \nabla f(x_k)^Td + \tfrac{1}{2} d^T \nabla_{xx}^2 \mathcal{L}(x_k,\lambda_k,\sigma_k) d \\
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| \mathrm{s.t.} & b(x_k) + \nabla b(x_k)^Td \ge 0 \\
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| & c(x_k) + \nabla c(x_k)^T d = 0. \end{array}</math>
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| Note that the term <math>f(x_k)</math> in the expression above may be left out for the minimization problem, since it is constant.
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| ==See also==
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| * [[Sequential linear programming]]
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| * [[Secant method]]
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| * [[Newton's method]]
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| ==References==
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| * {{cite book|last1=Bonnans|first1=J. Frédéric|last2=Gilbert|first2=J. Charles|last3=Lemaréchal|first3=Claude|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 <!-- ''Optimisation numérique: Aspects théoriques et pratiques'' --> 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|unused_data=<!-- authorlink3=Claude Lemaréchal -->}}
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| * {{ cite book | year=2006|url=http://www.ece.northwestern.edu/~nocedal/book/num-opt.html| title= Numerical Optimization| publisher=Springer.|isbn=0-387-30303-0| author=Jorge Nocedal and Stephen J. Wright}}
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| ==External links==
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| * [http://www.neos-guide.org/content/sequential-quadratic-programming Sequential Quadratic Programming at NEOS guide]
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| {{optimization algorithms}}
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| [[Category:Optimization algorithms and methods]]
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| {{Mathapplied-stub}}
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over the counter std test (official statement) writer is called Irwin. Puerto Rico is exactly where he's always been living but she requirements to move because of her family. Playing baseball is the pastime he will by no means quit performing. I am a meter reader.