chapter  11
40 Pages

Multivariate optimization

This chapter is concerned with multivariate optimization, finding values for (x1,

x2, . . . ,xn) to make the function f (x1,x2, . . . ,xn) as large or small as possible. In Sec-

tion 11.2, the two-variable case is described; Section 11.3 considers the

n-variable case. The extension from two to n variables is natural. Section 11.2 pro-

vides sufficient conditions for a local optimum and provides motivation for those

conditions. Also, in this section, the envelope theorem is described. Section 11.3

provides sufficient conditions for both local and global optima. These conditions

are developed in terms of concavity of the objective function and the properties of

Hessian matrices in Section 11.4.