Equality Constraints

Abstract

Let x = (x_l, ..., x_n) be an element of Eⁿ, the n dimensional Euclidian space, n < ∞, with the usual norm:$\left\| {\text{x}} \right\| = {({}_{\text{i}}\sum\limits_ = ^{\text{n}} {{}_1{\text{x}}_1^2} )^{1/2}}$. Let f(x) be defined on all of Eⁿ with values in R, denoting real numbers R = E¹. Let g(x) be defined on all of Eⁿ with values in E^m, i.e. ${\text{g}}({\text{x}}) = \left[ {\begin{array}{*{20}{c}}{{{\text{g}}^1}({\text{x}})} \\ \begin{gathered} {{\text{g}}^2}({\text{x}}) \hfill \\ \vdots \hfill \\ \end{gathered} \\ {{{\text{g}}^{\text{m}}}({\text{x}})} \end{array}} \right]$ and let g^α(x) denote the components of g(x) as α = 1, ..., m. Throughout this chapter, we shall assume that m is less than n.