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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: Inequality Constraints, Necessary Conditions, Sufficiency Conditions, Linear Constraints, Minimize, Continuously Differentiable, Activeinequality, Treated As Equations, Inactive Constraints, Lagrange Multipliers
Typology: Slides
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h = (h 1 , ..., hm), g = (g 1 , ..., gr ).
A(x) = j | gj (x) = 0.
m
i=
r
j=
∇f (x∗ ) + λ∗ i ∇hi(x μ∗ j ∇gj (x∗^ ) = 0,
∗ j = 0,^ ∀^ j /∈^ A(x μ ∗^ ).
∗ μ j = −(∆cost due to uj )/uj
g^ + j (x) = max^0 , gj^ (x)^ ,^ j^ = 1,... , r,
r
j=
λ^ ∗ i =^ lim^ khi(x
k (^) ), i = 1,... , m, k→∞
μ^ ∗ j =^ lim^ kgj
+(xk (^) ), j = 1,... , r. k→∞
j
j†x≤bj†, j=1,...,r^
μ^ ∗^ ∗^ ∗
r ∇f (x^ ∗ ) + μ∗^ ∗^ ∗ j aj^ = 0,^ μj = 0,^ ∀^ j /∈^ A(x^ ). j=
a 2
C⊥^ = {y | aj'y ≤ 0, j=1,...,r}
a 1
r
a 2
r
a 1
Cone generated by a (^) j , j ∈ A(x*^ )
− ∇f(x^ ) x
Constraint set
{x | aj'x ≤ bj, j = 1,...,r}
The local min x^ ∗ of the original problem is also a local min for the problem mina′ j†x≤bj†, j∈A(x∗^ )^
f (x). Hence
∇f (x^ ∗ )′(x − x∗ ) ≥ 0 , ∀ x with a ∗ j
′ (^) x ≤ bj , j ∈ A(x ).
Since a constraint a′ j x ≤ bj , j ∈ A(x∗) can also be ex-
pressed as a′ j (x − x∗) ≤ 0, we have
∇f (x^ ∗ )′^ y ≥ 0 , ∀ y with a ∗ j
′ (^) y ≤ 0 , j ∈ A(x ).
From Farkas’ lemma, −∇f (x∗) has the form
μ^ ∗^ ∗ j aj^ ,^ for some^ μj ≥^ 0,^ j^ ∈^ A(x
j∈A(x∗^ )
Let μ^ ∗ j = 0 for^ j /∈^ A(x
∗
∗ μ j ≥ 0 , j ∈ J,
∗
μ ∗^ ),
∗ (^) = arg min a′j†^ x≤bj j /∈J†
f (x) + j∈J
x μ ∗ j (a′j x − bj ).
