9.6 Constrained optimization and Lagrange multipliers

Main points:

• Constrained optimization
○ External circumstances constraining optimization
○ Graphical approach
§ Global max or min occurs where the graph of the constraint is tangent to a contour or at an endpoint of the constraint
○ Analytical approach (Lagrange multipliers)
§ F has a local maximum at P0 subject to the constraint if f(P0)≥f(P) for all point P near P0 satisfying the constraint
§ F has a global maximum at P0 subject to the constraint if f(P0)≥f(P) for all point P near P0 satisfying the constraint
§ Method of Lagrange Multipliers
fx(x,y) = λgx(x,y)
fy(x,y) = λgy(x,y)
g(x,y) = c
§ Lagrange multiplier ≈ delta f / delta g = change in optimum value of f / change in g
□ The value of Lagrange multiplier is approximately the change in the optimum value of when the value of the constraint is increased by 1 unit
□ The value of Lagrange multiplier represents the rate of change of the optimum value of f as the constraint increases
§ The Lagrangian fuction
□ L(x,y,λ) = f(x,y) - λ(g(x,y) - c)

Challenges:

I found the concept of Lagrange multiplier difficult and couldn’t really follow the explanation for it. It was also frustrating because there were several problems relating to economics. Similarly I had troubles understanding the Lagrangian function.

Reflection:

Constrained optimizations was interesting because both examples used in the book related to economics. Regardless that I haven't done either of the topics in the past the constrained optimization was a lot easier to understand compared to Lagrange multipliers. I'm hoping to really understand the Lagrange multiplier after tomorrow's class.