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| | My name is Santiago (49 years old) and my hobbies are Camping and Vintage car.<br><br>Visit my blog: [http://tinyurl.com/q6d8aya Cheap ghd uk] |
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| In [[mathematics]], '''Slater's condition''' (or '''Slater condition''') is a [[sufficient condition]] for [[strong duality]] to hold for a [[convex optimization|convex optimization problem]]. This is a specific example of a [[constraint qualification]]. In particular, if Slater's condition holds for the [[primal problem]], then the [[duality gap]] is 0, and if the dual value is finite then it is attained.<ref>{{cite book |last1=Borwein |first1=Jonathan |last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=978-0-387-29570-1}}</ref>
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| ==Mathematics==
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| Given the problem
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| :<math> \text{Minimize }\; f_0(x) </math>
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| :<math> \text{subject to: }\ </math>
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| ::<math> f_i(x) \le 0 , i = 1,\ldots,m</math>
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| ::<math> Ax = b</math>
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| with <math>f_0,\ldots,f_m</math> [[convex function|convex]] (and therefore a convex optimization problem). Then Slater's condition implies that strong duality holds if there exists an <math>x \in \operatorname{relint}(D)</math> (where relint is the [[relative interior]] and <math>D = \cap_{i = 0}^m \operatorname{dom}(f_i)</math>) such that
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| :<math>f_i(x) < 0, i = 1,\ldots,m</math> and
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| :<math>Ax = b.\,</math><ref name="boyd">{{cite book |last1=Boyd |first1=Stephen |last2=Vandenberghe |first2=Lieven |title=Convex Optimization |publisher=Cambridge University Press |year=2004 |isbn=978-0-521-83378-3 |url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf |format=pdf |accessdate=October 3, 2011}}</ref>
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| If the first <math>k</math> constraints, <math>f_1,\ldots,f_k</math> are [[linear function]]s, then strong duality holds if there exists an <math>x \in \operatorname{relint}(D)</math> such that
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| :<math>f_i(x) \le 0, i = 1,\ldots,k,</math>
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| :<math>f_i(x) < 0, i = k+1,\ldots,m,</math> and
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| :<math>Ax = b.\,</math><ref name="boyd" />
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| ===Generalized Inequalities===
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| Given the problem
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| :<math> \text{Minimize }\; f_0(x) </math>
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| :<math> \text{subject to: }\ </math>
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| ::<math> f_i(x) \le_{K_i} 0 , i = 1,\ldots,m</math>
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| ::<math> Ax = b</math>
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| where <math>f_0</math> is convex and <math>f_i</math> is <math>K_i</math>-convex for each <math>i</math>. Then Slater's condition says that if there exists an <math>x \in \operatorname{relint}(D)</math> such that
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| :<math>f_i(x) <_{K_i} 0, i = 1,\ldots,m</math> and | |
| :<math>Ax = b</math>
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| then strong duality holds.<ref name="boyd" />
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| ==References==
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| {{Reflist}}
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| [[Category:Mathematical optimization]]
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