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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: Optimality Conditions, Nonempty, Convex, Continuously Differentiable, Global Minima, Minima, Proposition, Local Minimum, Convex, Sufficient
Typology: Slides
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Problem: minx∈X†f (x),†where:
(a) X†⊂ n†^ is nonempty, convex, and closed. (b) f†is continuously differentiable over X.
f(x)
x
Local Minima (^) Global Minimum X Docsity.com
Proposition (Optimality Condition)
(a) If x†∗^ is a local minimum of f†over X, then
∇f (x†∗)′(x†− x†∗) ≥ 0 ,† ∀ x†∈ X.†
(b) If f†is convex over X, then this condition is also sufficient for x†∗^ to minimize f†over X.
Surfaces of equal cost f(x)
Constraint set X
x x*
∇f(x*)
Constraint set X
x* x
∇f(x*)
At a local minimum x†^ ∗ , the gradient ∇f†(x†∗) makes an angle less than or equal to 90 degrees with all fea- sible variations x−x†^ ∗ , x†∈ X.
Illustration of failure of the optimality condition when X† is not convex. Here x†^ ∗ is a local min but we have ∇f (x†∗)′(x†− x†∗) <† 0 for the feasible vector x†shown.
∑^ n
i=
∂f (x∗) ∂xi^ (xi^ −^ x
∗ i ) ≥ 0 , ∀ xi ≥ 0 , i = 1,... , n.
∂f (x∗) ∂xi
≥ 0 , ∀ i.
∂f (x∗) ∂xi
= 0, if x∗ i > 0.
x^ x = 0
∇f(x)^ ∇f(x)
x
∣ x^ ≥^0 ,
∑^ n
i=
xi = r
where r > 0 is a given scalar.
∑^ n
i=
∂f (x∗) ∂xi^ (xi−x
∗ i ) ≥ 0 , ∀ xi ≥ 0 with ∑^ n
i=
xi = r.
( ∂f (x∗) ∂xj
∂f (x∗) ∂xi
x∗ i ≥ 0 ,
x∗ i > 0 =⇒
∂f (x∗) ∂xi
∂f (x∗) ∂xj
, ∀ j.
flow of path p and let Tij
p: crossing (i,j) xp
be the travel time of link (i, j).
x∗ p > 0 =⇒ tp(x∗) ≤ tp′ (x∗), ∀ p′^ ∈ Pw , ∀ w ∈ W
where tp(x), is the travel time of path p
tp(x) =
all arcs (i,j) on path p
Tij (Fij ), ∀ p ∈ Pw , ∀ w ∈ W.
Identical with the optimality condition of the rout- ing problem if we identify the arc travel time Tij (Fij ) with the cost derivative D′ ij (Fij ).
minimize f (x) = ‖z − x‖^2 subject to x ∈ X.
Proposition (Projection Theorem) Problem has a unique solution [z]+^ (the projection of z).
z
x
Constraint set X
x *
x - x* z - x*
Necessary and sufficient con- dition for x∗^ to be the pro- jection. The angle between z − x∗^ and x − x∗^ should be greater or equal to 90 degrees for all x ∈ X, or (z − x∗)′(x − x∗) ≤ 0
‖[x]+^ − [y]+‖ ≤ ‖x − y‖, ∀ x, y ∈ n.
