# Calculus Optimization Methods

A key application of calculus is in optimization: finding maximum and minimum values of a function, and which points realize these extrema.

## Contents

### Context

Formally, the field of mathematical optimization is called mathematical programming, and calculus methods of optimization are basic forms of nonlinear programming. We will primarily discuss finite-dimensional optimization, illustrating with functions in 1 or 2 variables, and algebraically discussing n variables. We will also indicate some extensions to infinite-dimensional optimization, such as calculus of variations, which is a primary application of these methods in physics.

### Techniques

Basic techniques include the first and second derivative test, and their higher-dimensional generalizations.

A more advanced technique is Lagrange multipliers, and generalizations as Karush–Kuhn–Tucker conditions and Lagrange multipliers on Banach spaces.

### Applications

Optimization, particularly via Lagrange multipliers, is particularly used in the following fields:

Further, several areas of mathematics can be understood as generalizations of these methods, notably Morse theory and calculus of variations.

### Terminology

• Input points, output values
• Maxima, minima, extrema, optima
• Stationary point, critical point; stationary value, critical value
• Objective function
• Constraints – equality and inequality
• Especially sublevel sets
• Feasible region, whose points are candidate solutions

## Statement

This tutorial presents an introduction to optimization problems that involve finding a maximum or a minimum value of an objective function $f(x_1,x_2,\ldots, x_n)$ subject to a constraint of the form $g(x_1,x_2,\ldots, x_n)=k$.

## Maximum and minimum

Finding optimum values of the function $f(x_1,x_2,\ldots, x_n)$ without a constraint is a well known problem dealt with in calculus courses. One would normally use the gradient to find stationary points. Then check all stationary and boundary points to find optimum values.

### Example

• $f(x,y)=2x^2+y^2$
• $f_x(x,y)=4x=0$
• $f_y(x,y)=2y=0$

$f(x,y)$ has one stationary point at (0,0).

## The Hessian

A common method of determining whether or not a function has an extreme value at a stationary point is to evaluate the hessian of the function at that point. where the hessian is defined as

$H(f)= \begin{bmatrix} \frac{{\partial}^2 f}{\partial x_1^2} & \frac{{\partial}^2 f}{\partial x_1 \partial x_2} & \dots & \frac{{\partial}^2 f}{\partial x_1 \partial x_n} \\ \frac{{\partial}^2 f}{\partial x_2 \partial x_1} & \frac{{\partial}^2f}{\partial x_2^2}& \dots & \frac{{\partial}^2f}{\partial x_2 \partial x_n}\\ \vdots & \vdots & \ddots & \vdots \\ \frac{{\partial}^2f}{\partial x_n \partial x_1} & \frac{{\partial}^2f}{\partial x_n \partial x_2}& \dots & \frac{{\partial}^2f}{\partial x_n^2}\\ \end{bmatrix}.$

## Second derivative test

The Second derivative test determines the optimality of stationary point $x$ according to the following rules [2]:

• If $H(f)>0$ at point x then $f$ has a local minimum at x
• If $H(f) < 0$ at point x then $f$ has a local maximum at x
• If $H(f)$ has negative and positive eigenvalues then x is a saddle point
• Otherwise the test is inconclusive

In the above example.

$H(f)=\begin{bmatrix} 4 & 0\\ 0& 2 \end{bmatrix}.$

Therefore $f(x,y)$ has a minimum at (0,0).

## References

[1] T.K. Moon and W.C. Stirling. Mathematical Methods and Algorithms for Signal Processing. Prentice Hall. 2000.
[2]http://www.ece.tamu.edu/~chmbrlnd/Courses/ECEN601/ECEN601-Chap3.pdf