CONSTRAINED MINIMIZATION

CONSTRAINED MINIMIZATION

AIM

Minimization of a given function subjected to constraints f(x,y)

OBJECTIVE

Minimize the following function

f(x)=5-(x-2)^2 -2(y-1)^2

subject to the following constraint

x + 4y = 3

DESCRIPTION

To find a local minimum or stationary point of f(x,y)

Subject to the equality constraints g(x,y)

Where the line is the constraint line g(x,y)=0 and (a,b) will be the desired minimum point.

The bowl function is the objective function, to be minimized. The plane represents the constraint. When we limit or constrain the search for the minimum only to points that satisfy the constraint, by intuition we will get the indicated point.

The minimum of F(x,y) along g(x,y)=0 at (a,b) is just the point where the constraint line is tangent to a level curve or contour line. It is also at the point where the constraint line is perpendicular to the direction of the gradient, or the direction of maximum slope. The gradient at a point is,

This gives the direction in which the rate of change of the function value is maximum. A line in the plane, through (a,b) and parallel to the gradient is,

A line through (a,b) perpendicular to the gradient, or tangent to the level curve is

At the desired minimum point, the line tangent to the level curve must be coincident with the constraint line

If we take the two equations in (9), plus the constraint equation (2) together

We now have 3 equations in 3 unknowns, (a,b,)

Which we can solve. We are generally not interested in the explicit value for lambda, it is necessary but only an intermediate step. So we solve the equations, keeping the (a,b) and throwing away the lambda, the Lagrange Multiplier.

EQUATIONS

PROBLEM

RESULT

Minimization of a function using constraints completed and output verified