A Course on Statistics for Finance

Appendix C

Lagrange Multipliers

C.1 Notation

(See also Appendix A on Vectors and Matrices.)

The symbol x denotes an n -dimensional vector, with elements x 1 , x 2 , . . . , x n .

The functions f and g are scalar functions of x. The dot product (inner product , scalar product) of vectors u and v is denoted by u · v. (See Appendix A - Vectors and Matrices.)

The null vector is denoted by 0. The gradient of f (x), denoted by grad f (x), is the vector of partial derivatives of f with respect to the elements of x :

$\text{grad}f=(\partial f/\partial x_1\partial f/\partial x_2\mathrm{...}\partial f/\partial x_n).\text{ }$

C.2 Optimization Problem

The following general optimization problem is considered.

$\text{Maximize }f\text{(}x\text{), subject to }x\text{ in }S\text{, where }S\text{ = {}x\text{ : }g\text{(}x\text{) = 0}}\text{.}$

Remarks. (i) This is called “maximizing ...