Lagrange Multiplier

Its a method of Solving optimization problem over Real field with respect to equality constraints.

Let, the problem P is :-

Find max f(x,y)

s.t.  g(x,y) = 0

Now,

1. We will make another function  φ(x,y,λ) = f(x,y) + λ . g(x,y).

2. We can say that  ∀p = (x₀, y₀) ∈ sp(P) [ sp(P) is Solution Space of problem P ] ∃λ₀ s.t. φ(x₀, y₀, λ₀) will be a stationary point.

i.e.  ∀(x₀, y₀) ∈ sp(P) ∃λ₀ ∈ ℜ s.t. φ(x₀, y₀, λ₀) will be a stationary point,
→  ∇_x,y,λ (φ) | _{(x0, y0, λ0)} = 0.

One Example :-

max ( x + y )
s.t. x² + y² = 1

φ(x, y, λ) = x + y + λ ( x² + y² - 1 )
∇_x,y,λ (φ) = ( 1 + λ x ; 1 + λ x ; x² + y² - 1) = 0 = ( 0 , 0 , 0 ) 
→ λ = √ 2 , ( x , y ) =  ( 1/√ 2 ,1/√ 2 )