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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: Additional Methods, Least-Squares Problems, Incremental, Conjugate Direction Methods, Conjugate Gradient Method, Quasi-Newton Methods, Coordinate Descent Methods, Recall the Least-Squares Problem, Minimize, Steepest Descent Method
Typology: Slides
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minimize f (x) = 12 ‖g(x)‖^2 = 21
m
i=
‖gi(x)‖^2
subject to x ∈ n^ ,
where g = (g 1 ,... , gm), gi : n^ → ri†.
xk+1^ = xk^ −αk∇f (xk) = xk^ −αk
m
i=
∇gi(xk)gi(xk)
ψi = ψi− 1 − αk∇gi(ψi− 1 )gi(ψi− 1 ), i = 1,... , m
ψ 0 = xk^ , xk+1^ = ψm
(ai x - bi )^2
x* mini^ a bii Advantage of incrementalism
R (^) maxi abi x i
y^0 y^1 y^2 w 0
w^1
x^0 x^1 x^2
d 0 = Q -1/2^ w^0
d^1 = Q -1/2w^1 Expanding Subspace Theorem
di+1^ = ξi+1^ +
i
m=
c(i+1)mdm;
choose c(i+1)m^ so di+1^ is Q-conjugate to d^0 ,... , di,
Qdj^ = 0.
i
m=
di+1′Qdj^ = ξi+1′Qdj^ + c(i+1)mdm
d^2 = ξ^2 + c^20 d^0 + c^21 d^1 d^1 = ξ^1 + c^10 d^0 ξ 2
d 1 d 0
ξ^1
0 (^0) - c (^10) d (^0) ξ (^0) = d 0
qk^ ≈ ∇^2 f (xk+1)pk,
pk^ = xk+1^ − xk, qk^ = ∇f (xk+1) − ∇f (xk). ]− 1 ∇^2 f (xn) ≈ q^0 · · · qn−^1 p^0 · · · pn−^1
Dk+1^ = Dk^ +
pkpk′ −
Dkqkqk′Dk
vk^ =
pk −
Dkqk , τ k^ = qk′Dkqk^ , 0 ≤ ξk^ ≤ 1 pk′^ qk^ τ^ k
and D^0 > 0 is arbitrary, αk^ by line minimization, and Dn^ = Q−^1 for a quadratic.
∂f (xk) ≈
f (xk^ + hei) − f (xk) ∂xi^ h
∂f (xk) ≈
f (xk^ + hei) − f (xk^ − hei) ∂xi^2 h
xk
x k+1^ x
k+
