The Maximization of a Quadratic Function of Variables Subject to Linear Inequalities
A simplex-type method for finding a local maximum of <disp-formula><tex-math><![CDATA[$$Z = c + c\lambda^{\prime} + \lambda C\lambda^{\prime} \qquad a\ \max \eqno (1)$$]]></tex-math></disp-formula> subject to <disp-formula><tex-math><![CDATA[$$\lambda \geqq 0 \eqno (2)$$]]></tex-math></disp-formula> and <disp-formula><tex-math><![CDATA[$$A\lambda^{\prime} = \hbox{b}^{\prime} \eqno (3)$$]]></tex-math></disp-formula> is proposed. At a local maximum, the objective function (1), can be expressed, in terms of the non-basic variables \lambda<sub>0</sub>, as <disp-formula><tex-math><![CDATA[$$Z = \alpha + \beta\lambda_0^{\prime} + \lambda_0\gamma\lambda_0^{\prime} \eqno (13)$$]]></tex-math></disp-formula> and the vector of partial derivatives of (13), with respect to the non-basic variables may be written, <disp-formula><tex-math><![CDATA[$$\nabla {\bf Z} = \beta + 2\lambda_0\gamma;\qquad \beta \geqq 0$$]]></tex-math></disp-formula> This allows calculation of the maximum values of the non-basic variables, increased one at a time, consistent with \nabla Z \geqq 0. A "cutting plane" a' \lambda' \geqq 1 is then defined which excludes the local optimum, and many lower values (but no higher values) of (1). The form of the square matrix C is immaterial.
