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These are the Lecture Slides of Nonlinear Programming which includes Convex Cost, Linear Constraints, Duality Theorem, Linear Programming Duality, Quadratic Programming Duality, Linear Inequality, Constrained Problem, Minimize, Feasible etc.Key important points are: Interior Point Methods, Barrier, Linear Programs, Logarithmic Barrier, Inequality Constrained Problem, Minimize, Continuous, Set, Feasiblepoint, Approaches
Typology: Slides
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S = x ∈ X | gj (x) < 0 , j = 1,... , r
r r 1 B(x) = − ln −gj (x) , B(x) = −. gj (x) j=1 j=
xk^ = arg min f (x) + �k^ B(x) , k = 0, 1 ,... , x∈S
Boundary of S Boundary of S
ε B(x)
ε' B(x)
ε' < ε
S
c ,
n
i=
x(�) = arg min^ ′^ x − � x∈S
F�(x) = arg min x∈S
ln xi
All central paths start at Point x(ε) on central path
x∞
S
x* (ε = 0)
c
and end at optimal solu tions of (LP).
the analytic center
n
i=
x∞ = arg min x∈S
ln xi
x ˜ = x + α(x − x),
Az=b
x = x − Xq(x, �),
Xz q(x, �) = − e, e = (1... 1)′^ , z = c − A′λ,
λ = (AX^2 A′)−^1 AX Xc − �e ,
Following approximately the
Central Path central path by using a sin
x
Set {x | ||q(x,εk^ )|| < 1}
x∞
x(εk+1^ ) xk x(εk^ )
xk+
Set {x | ||q(x,εk+1^ )|| < 1}
S
gle Newton step for each �k^. If �k^ is close to �k+ and xk^ is close to the cen tral path, one expects that xk+1^ obtained from xk^ by a single pure Newton step will also be close to the central path.
‖q(x, �)‖ ≤
γ^2 + δ 1 − δn−^1 /2
δ ≤ γ(1 − γ)(1 + γ)−^1 ,
x^ * * Central Path
x∞
x(εk+1^ ) x k x(εk^ )
xk+
x k+2 x(εk+2^ )
x
S
Central Path
x∞
x(εk+1^ ) xk x(εk^ )
xk+
x (^) x(εk+2 (^) ) k+
S
(a) (b)
