How do you minimize and maximize f(x,y)=x^2+y^3f(x,y)=x2+y3 constrained to 0<x+3xy<40<x+3xy<4?

1 Answer
Jun 15, 2016

A local minimum at {x= 1.12872, y= 0.847942}{x=1.12872,y=0.847942} and
a local minimum also at {x = 0, y = 0}{x=0,y=0}

Explanation:

Introducing the so called slack variables s_1, s_2s1,s2 the optimization problem is transformed into an equivalent one

Find local minima, maxima of

f(x,y)= x^2+y^3f(x,y)=x2+y3

subject to

g_1(x,y,s_1) = x + 3 x y - s_1^2 = 0g1(x,y,s1)=x+3xys21=0
g_2(x.y,s_2)=x+3 x y + s_2^2-4=0g2(x.y,s2)=x+3xy+s224=0

The lagrangian is

L(x,y,s_1,s_2,lambda_1,lambda_2) = f(x,y)+lambda_1 g_1(x,y,s_1)+lambda_2g_2(x,y,s_2)L(x,y,s1,s2,λ1,λ2)=f(x,y)+λ1g1(x,y,s1)+λ2g2(x,y,s2)

LL is analytical so the stationary points include the relative maxima and minima.

The determination of stationary points is done solving for x,y,s_1,lambda_1,s_2,lambda_2x,y,s1,λ1,s2,λ2 the system of equations given by

grad L(x,y,s_1,lambda_1,s_2,lambda_2) = vec 0L(x,y,s1,λ1,s2,λ2)=0

or

{(2 x + lambda_1 (1 + 3 y) + lambda_2 (1 + 3 y)=0), (3 lambda_1 x + 3 lambda_2 x + 3 y^2=0), ( -s_1^2 + x + 3 x y=0), (-2 lambda_1 s_1=0), (-4 + s_2^2 + x + 3 x y=0) ,( 2 lambda_2 s_2=0) :}

Solving we get

(x=0., y = 0., lambda_1 = 0., s_1 = 0., lambda_2= 0., s_2= 2.)

and

(x= 1.12872, y= 0.847942, lambda_1=0., s_1 = 2., lambda_2 = -0.637008, s_2 = 0.)

Both points are at the boundaries of g_1(x,y,0)=0 and g_2(x,y,0)=0 respectively.

Their qualification must be done with f_(g_1) and f_{g_2} respectively. So,

f_{g_1}(x) = x^2-1/27
f_{g_2}(x) = -(x-4)^3/(27 x^3) + x^2

d^2/(dx^2)f_{g_1}(0)=2 qualifying this point as a local minimum
d^2/(dx^2)f_{g_2}(1.12872) =11.5726 qualifying this point as a local minimum also

Attached a figure with a contour mapping with the found points

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