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An introduction to duality theory in nonlinear programming, covering the geometrical framework, lagrange multipliers, and the dual problem. It includes examples of problems where a lagrange multiplier exists and does not exist, as well as the weak duality theorem and the relationship between the optimal primal and dual values.
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−∞ < f ∗^ = inf x∈X† gj†(x)≤ 0 , j=1,...,r†
f (x) < ∞
0 0
Min Common Point
Max Intercept Point Max Intercept Point
Min Common Point S S
(a) (b)
0 (0,-1)
(μ*,1) (0,1)
(a)
(-1,0)
S = {(g(x),f(x)) | x ∈ X}
(-1,0)^0
(μ*,1)
(b)
S = {(g(x),f(x)) | x ∈ X}
0
(μ*,1)
(c)
(μ*,1)
(μ*,1)
S = {(g(x),f(x)) | x ∈ X}
min f(x) = x 1 - x 2 s.t. g(x) = x 1 + x 2 - 1 ≤ 0 x ∈ X = {(x 1 ,x 2 ) | x 1 ≥ 0, x 2 ≥ 0 }
min f(x) = (1/2) (x 12 + x 22 ) s.t. g(x) = x 1 - 1 ≤ 0 x ∈ X = R^2
min f(x) = |x 1 | + x 2 s.t. g(x) = x 1 ≤ 0 x ∈ X = {(x 1 ,x 2 ) | x 2 ≥ 0 }
(0,f*) = (0,0)
S = {(g(x),f(x)) | x ∈ X} min f(x) = x s.t. g(x) = x^2 ≤ 0 x ∈ X = R
(a)
(-1/2,0)
S = {(g(x),f(x)) | x ∈ X}
(b)
(0,f*) = (0,0)
(1/2,-1)
min f(x) = - x s.t. g(x) = x - 1/2 ≤ 0 x ∈ X = {0,1}
x^ ∗ = arg min L(x, μ ∗^ ), μ∗^ ∗ x∈X j^ gj^ (x^ ) = 0,^ j^ = 1,... , r
Dq = μ | q(μ) > −∞.
q ∗^ ≤ f ∗.
r q(μ) = inf L(z, μ) ≤ f (x) + μj gj (x) ≤ f (x), z∈X j=
q ∗^ = sup q(μ) ≤ inf f (x) = f ∗. μ≥ 0 x∈X, g(x)≤^0
μ
q(μ)
f* = 0
(a)
f* = 0 1 μ
q(μ)
min f(x) = x s.t. g(x) = x^2 ≤ 0 x ∈ X = R
min f(x) = - x s.t. g(x) = x - 1/2 ≤ 0 x ∈ X = {0,1} q(μ) = min { - x + μ(x - 1/2)} = min{ - μ/2, μ/2 −1} x ∈ {0,1} (b)
