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lecture15

# lecture15 - Variational Inequality Do you remember Nicholas...

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IE417: Nonlinear Programming: Lecture 15 Jeﬀ Linderoth Department of Industrial and Systems Engineering Lehigh University 28th March 2006 Jeﬀ Linderoth IE417:Lecture 15 Variational Inequality? Do you remember Nicholas Steir’s Talk? He used First Order Conditions ! He also mentioned variational inequalities . Let Ω R n be a closed convex set, and consider the function/mapping f : Ω R n . The variational inequality problem involves ﬁnding a vector x * R n such that F ( x * ) T ( x - x * ) 0 x Ω What does this have to do with optimization? If x * is a local minimizer of f in Ω , then x * solves the following VIP: f ( x * ) T ( x - x * ) 0 x Ω Recall normal cone, for x Ω N Ω ( x ) = { v | v T ( w - x ) 0 w Ω } Geometrically, you want that -∇ f ( x ) N ( x ) Jeﬀ Linderoth IE417:Lecture 15 Loose Categories min x R n f ( x ) subject to c i ( x ) = 0 i ∈ E , c i ( x ) 0 i ∈ I 1 Quadratic Programming: f ( x ) def = x T Qx + d T i x + b c i ( x ) def = a T i x + b i i ∈ E ∪ I 2 Bound Constraint Optimization: c i ( x ) def = x i + b i i ∈ I , ( E = ) 3 Convex Programming: f ( x ) is convex, c i ( x ) is linear for all i ∈ E , and c i ( x ) is concave for all i ∈ I Jeﬀ Linderoth IE417:Lecture 15 Loose Categories: Solution Methods 1 Penalty Function: (Say for I = ) f ( x ) + 1 2 μ ± i ∈E c 2 i ( x ) 2 Barrier Method: (Say for E = ) f ( x ) + μ ± i ∈I log c i ( x ) 3 Augmented Lagrangian: (Say for I = ) L A ( x, λ, μ ) = f ( x ) - ± i ∈E λ i c i ( x ) + 1 2 μ ± i ∈E c 2 i ( x ) 4 Sequential Quadratic Programming. Iterate via solving a sequence of quadratic approximations to the Lagrangian that satisfy linear approximation to constraints Jeﬀ Linderoth IE417:Lecture 15

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lecture15 - Variational Inequality Do you remember Nicholas...

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