Tues. 10/28 Constrained optimization and Lagrange multipliers

9.6


1) Main Points

Chapter 9.6 introduces a more real world application of optimization that deals with constraints. An example of a constraint would be a budget, with constrained optimization you are looking the optimum (max or min) out put of a function within the limitations a budget. Constraints can also be graphed. If f(x,y) has a global max/min on the constraint g(x,y)=c, it occurs at the point where the graph of the constraint is tangent to a contour of f, or at an endpoint of the constraint. The method of Lagrange multiplies gives a system of three equations and states that "if f has a constrained global maximum or minimum, then it occurs at one of the solutions (x0,y0) to this system or at an endpoint of the constraint. This basically means that to find the min or max given constraints you have to differentiate fy and fx along with the equation for the constraint and solve these as a system of equations. The lagrange multiplier (the little tepee symbol) is approximately the change in the optimum value of f when the value of the constraint is increased by 1 unit. It represents the rate of change of the optimum value of f as the constraint (budget) increases. The lagrange multiplier can also be used to calculate the price the product must be sold for when the budget is increased by $1. The lagrange multiplier is also incorporated into the Lagrangian function in which critical points of the multivariable function can be found.


2) Challenges

I am confused about how to find the endpoints of a function. Constraints add a new step to optimization and involve more computation. Because there are two multivariable equations involved in unconstrained optimization it is easy to mix up what means what. Dealing with the Lagrange multiplier in general is difficult, it is important to be familiar with what it means.


3) Reflections

Constrained optimization seems to have much more real world application than unconstrained. This concept seems like something a business would actually use to calculate optimal production. I find the examples most understandable when they deal with money and production because they are actually used for those applications.