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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: Introduction to Duality, Convex Cost, Linear Constraints, Duality Theorem, Linear Programming Duality, Quadratic Programming Duality, Linear Inequality, Constrained Problem, Minimize, Feasible
Typology: Slides
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j ′ x ≤ bj , j = 1,... , r,
n
∗
∗
∗
j
∗
j
∗
∗ ∗ ′ x = arg min f (x) + μj (aj x − bj ). a′^ x≤bj j j / † j∈J ∈J†
∗
∗ ∗ μ j
j (a ′ ∗
j x r (^) ∗ ∇f (x ∗ ) + j= μ j
r ∗ ∗ ′ x = arg min f (x) + μj (aj x − bj ). x∈n† j= ∗
j (a ′ j x ∗
r ∗ ∗ ′ f (x ) = min f (x) + μj (aj x − bj ). x∈n† j= ∗
j (a ′ j
j
′ r ∗ ∗ ′ f (x ) ≤ min f (x) + μj (aj x − bj ) a′^ x≤bj j j / † j= ∈J† ∗ ′ ≤ min f (x) + μj (aj x − bj ). a′^ x≤bj j j / † j∈J ∈J†
′
min f (x) x∈X, a′ j x≤bj , j=1,...,r
n
r → [−∞, ∞) r j= q(μ) = inf f (x) + μj (aj x − bj ) x∈X L(x, μ) = inf x∈X
max μ≥ 0 q(μ).
Q = μ | μ ≥ 0 , q(μ) > −∞.
′
∗
∗
∗
∗
∗ ∗ ∗ ∗ f (x ) = L(x , μ ) = min x∈X L(x, μ ).
∗
′ j x − j (a ′
r j= f (x) + μj (a j q(μ) ≤ inf x − bj ) x∈X, a′ j x≤bj , j=1,...,r ∗ ≤ inf ). x∈X, a′ j x≤bj , j=1,...,r f (x) = f (x (*) ∗
∗
j (a ′ j x ∗ −
∗ = arg minx∈X L(x, μ ∗
∗ ) + r j= q(μ μ ∗ ) = L(x ∗ ∗ ) = f (x ∗ ′ j (aj x ∗ − bj ) = f (x ∗ , μ ).
′ x
′ x = di, i = 1,... , m, x ≥ 0
n m m q(λ) = inf cj − λieij xj + λidi. x≥ 0 j=1 i=1 i= m
i=
m
i= m
i=
m
i= m
i=
1
2 x ′ Qx + c ′ x
r
c ∈ n
q(μ) = inf x∈n 1 2 x ′ Qx + c ′ x + μ ′ (Ax − b).
− 1 (c + A ′
2 μ ′ AQ − 1 A ′ μ − μ ′ (b + AQ − 1 c) − 1 q(μ) = − 1 2 c ′ Q − 1 c.
2 μ ′ P μ + t ′ μ
− 1 A ′
− 1 c.
