Mathematics for Economists
Chapter 10 : Optimization, several variables
By Lund University
In the final chapter of this course we look at maximization and minimization of functions of several variables. Optimization can be either unconstrained or constrained. By constrained optimization, we mean that the variables must satisfy one or several constraints, we are not free to pick any values when we optimize the function. Unconstrained optimization is very similar to optimization of a function of a single variable. For constrained optimization, we will study a solution technique called the method of Lagrange.
Optimization, several variables
In the first section of this final chapter we look at optimization of functions of several variables. In this section we only look at unconstrained optimization. We will look at necessary and sufficient conditions for a local optimum point and we will look at the extreme value theorem and optimization of convex and concave functions. The optimization problem is very similar to optimization of a function of one variable, replacing ordinary derivatives with partial derivatives and second order derivatives with the Hessian.
Optimizing a function of 2 variables, necessary conditions
Optimizing a function of 2 variables, sufficient conditions
Optimizing a function of 2 variables, additional results
Multivariable optimization, n variables
Optimization: Problems
Exercises on multivariable optimization
Problem: Maximize function of two variables
Problem: Minimize function of two variables
Problem: Maximize profit function
Problem: Second derivative test
Problem: Maximize utility function
Problem: Maximize profit function
Problem: First- and second derivative test
Problem: Classify stationary points
Problem: Find extreme points
Problem: Find maximum and minimum points
Problem: Optimizing function of 2 variables
Problem: open, closed, bounded and compact sets
Constrained optimization
In the final theory section of this course we will look at constrained optimization. Specifically, we will introduce something called the Lagrangian and the Lagrange multiplier and we will study the method of Lagrange. This method will help us to find the optimal point that at the same time satisfy our constraint.
Constrained optimization
Constrained optimization: Problems
Exercises on constrained optimization