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Linear Programming Non-Linear Optimization, Lecture Notes - Mathematics - Prof. J Vanderbei.pdf, Prof. J Vanderbei, Mathematics, Linear Programming, Non-Linear Optimization, The Interior-Point Algorithm, Reduced KKT System, Convex Optimization Models, FIR Filter Design, [Woofer, Midrange, Tweeter], Celestial Mechanics, Goddard Rocket Problem
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Robert J. Vanderbei
December 12, 2005 ORF 522
Operations Research and Financial Engineering, Princeton University http://www.princeton.edu/∼rvdb
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minimize f (x) subject to hi(x) ≥ 0 , i = 1,... , m
minimize f (x) subject to h(x) − w = 0, w ≥ 0
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minimize f (x) − μ
∑^ m
i=
log(wi)
subject to h(x) − w = 0
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∇f (x) − ∇h(x)T^ y = 0 W Y e = μe h(x) − w = 0
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H(x, y) 0 −A(x)T 0 Y W A(x) −I 0
∆x ∆w ∆y
−∇f (x) + A(x)T^ y μe − W Y e −h(x) + w
Here,
H(x, y) = ∇^2 f (x) −
∑^ m
i=
yi∇^2 hi(x)
and A(x) = ∇h(x)
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Nonlinear Programming (NLP)
minimize f (x) subject to hi(x) = 0, i ∈ E, hi(x) ≥ 0 , i ∈ I. NLP is convex if
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For convex nonquadratic optimization, it does not suffice to choose the steplength α simply to maintain positivity of nonnegative vari- ables.
f (x) = (1 + x^2 )^1 /^2.
x(k+1)^ = −(x(k))^3
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Notation: let N˜ denote the dual normal matrix associated with H˜.
Theorem If N˜ is positive definite, then (∆x, ∆w, ∆y) is a descent direction for
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∆x ∆y
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Scherk.mod with D discretized into a 64 × 64 grid gives the following results: constraints 0 variables 3844 time (secs) loqo 5. lancelot 4. snopt *
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