2.3 Mathematical Tools
59
The last equality determines that λ = 0 whenever g(x) < 0, i.e. when the constrained is not active, and g(x) = 0 when λ > 0, i.e. when the constraint is active.
The special (neutral) case that λ = 0 and g(x) = 0 hold simultaneously is obtained
when F(x) attains its minimum for the same x 0 that also denotes the root of g(x) = 0.
Alternatively, reformulating the inequality constraint into a corresponding (nonsmooth) equality constraint g(x) = 0, the Lagrange multiplier method is expressed
in terms of a modified Lagrange function (functional) ˆ
(x, λ)
ˆ
(x, λ) := F(x) + λ g(x).
(2.133)
Consequently, the optimality condition then reads formally
F
(x) + λ g(x)
0
(2.134)
which has to be solved jointly with the constraint condition
g(x) = 0.
(2.135)
Of course, the non-smoothness of the constraint g(x) may pose difficulties upon
computing its derivative (indeed its sub-differential, i.e. the set of sub-derivatives)
with respect to x at x = x 0 . Nevertheless, formally, the use of g(x) as an equality
constraint alleviates the representation of the following variants of the Lagrange
multiplier method.
Perturbed Lagrange Multiplier Method
For the perturbed Lagrange multiplier method the function (functional) F(x) to be
minimized (under the restriction of the alternative equality constraint g(x) = 0) is
amended by the product of the positive Lagrange multiplier λ ≥ 0 and the alternative
equality constraint g(x) = 0 together with the weighted square −1/[2 ε] λ
2
≤ 0
of the Lagrange multiplier to render the perturbed Lagrange function (functional)
ˆ
ε (x, λ)
ˆ
ε (x, λ) := F(x) + λ
g(x) −
1
2ε
λ
.
(2.136)
Correspondingly, the optimality condition then reads
F
(x) + λ g(x)
0
(2.137)
which has to be solved jointly with the constraint condition
λ = ε g(x) ≥ 0.
(2.138)
59
The last equality determines that λ = 0 whenever g(x) < 0, i.e. when the constrained is not active, and g(x) = 0 when λ > 0, i.e. when the constraint is active.
The special (neutral) case that λ = 0 and g(x) = 0 hold simultaneously is obtained
when F(x) attains its minimum for the same x 0 that also denotes the root of g(x) = 0.
Alternatively, reformulating the inequality constraint into a corresponding (nonsmooth) equality constraint g(x) = 0, the Lagrange multiplier method is expressed
in terms of a modified Lagrange function (functional) ˆ
(x, λ)
ˆ
(x, λ) := F(x) + λ g(x).
(2.133)
Consequently, the optimality condition then reads formally
F
(x) + λ g(x)
0
(2.134)
which has to be solved jointly with the constraint condition
g(x) = 0.
(2.135)
Of course, the non-smoothness of the constraint g(x) may pose difficulties upon
computing its derivative (indeed its sub-differential, i.e. the set of sub-derivatives)
with respect to x at x = x 0 . Nevertheless, formally, the use of g(x) as an equality
constraint alleviates the representation of the following variants of the Lagrange
multiplier method.
Perturbed Lagrange Multiplier Method
For the perturbed Lagrange multiplier method the function (functional) F(x) to be
minimized (under the restriction of the alternative equality constraint g(x) = 0) is
amended by the product of the positive Lagrange multiplier λ ≥ 0 and the alternative
equality constraint g(x) = 0 together with the weighted square −1/[2 ε] λ
2
≤ 0
of the Lagrange multiplier to render the perturbed Lagrange function (functional)
ˆ
ε (x, λ)
ˆ
ε (x, λ) := F(x) + λ
g(x) −
1
2ε
λ
.
(2.136)
Correspondingly, the optimality condition then reads
F
(x) + λ g(x)
0
(2.137)
which has to be solved jointly with the constraint condition
λ = ε g(x) ≥ 0.
(2.138)