lecture17

lecture17 - Penalty Methods Focus first on equality...

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IE417: Nonlinear Programming: Lecture 17 Jeﬀ Linderoth Department of Industrial and Systems Engineering Lehigh University 6th April 2006 Jeﬀ Linderoth IE417:Lecture 17 Penalty Methods Focus ﬁrst on equality constrained problem: min f ( x ) s.t. c i ( x ) = 0 i ∈ E (ENLP) Consider the quadratic penalty function: Q ( x, μ ) def = f ( x ) + μ 2 ± i ∈E c i ( x ) 2 and the unconstrained problem min x Q ( x, ˆ μ ) for a ﬁxed ˆ μ . As μ → ∞ , then minimizer of Q ( x, μ ) must have c i ( x ) = 0 i ∈ E Idea: solve ENLP by solving a sequence of unconstrained problems Theorem Let x k arg min Q ( x, μ k ) be an exact global minimizer, and let μ k → ∞ . Every limit point x * of the sequence { x k } solves ENLP. Jeﬀ Linderoth IE417:Lecture 17 Approximate Solves This is of little practical value, since ﬁnding a global minimizer for each Q ( x, μ k ) is likely intractable. Suppose instead that we ﬁnd x k such that ± Q ( x k , μ k ) ± ≤ τ k At each iteration, we ﬁnd an approximate (ﬁrst-order) minimizer. Theorem Let the limits lim k →∞ τ k = 0 lim k →∞ μ k → ∞ . Then for all limit points x * of the sequences { x k } at which the constraint gradients are independent, x * is a KKT point for ENLP. Further the multipliers will satisfy the following relationship: lim k →∞ - μ k c i ( x k ) = λ * i i ∈ E Let’s do the proof?

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This note was uploaded on 02/29/2008 for the course IE 417 taught by Professor Linderoth during the Spring '08 term at Lehigh University .

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lecture17 - Penalty Methods Focus first on equality...

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