Chapter 8: Q2E (page 437)

Question 2: Given points \({{\mathop{\rm p}\nolimits} _1} = \left( {\begin{array}{{20}{c}}0\\{ - 1}\end{array}} \right),{\rm{ }}{{\mathop{\rm p}\nolimits} _2} = \left( {\begin{array}{{20}{c}}2\\1\end{array}} \right),\) and \({{\mathop{\rm p}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}1\\2\end{array}} \right)\) in \({\mathbb{R}^{\bf{2}}}\), let \(S = {\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3}} \right\}\). For each linear functional \(f\), find the maximum value \(m\) of \(f\), find the maximum value \(m\) of \(f\) on the set \(S\), and find all points x in \(S\) at which \(f\left( {\mathop{\rm x}\nolimits} \right) = m\).
a. \(f\left( {{x_1},{x_2}} \right) = {x_1} + {x_2}\)
b. \(f\left( {{x_1},{x_2}} \right) = {x_1} - {x_2}\)
c. \(f\left( {{x_1},{x_2}} \right) = - 2{x_1} + {x_2}\)

Short Answer

Expert verified

On the set conv\(\left\{ {{{\bf{p}}_2},{{\bf{p}}_3}} \right\}\) is the point in \(S\) at \(m = 3\).
On the set \({\rm{conv}}\left\{ {{{\bf{p}}_1},{{\bf{p}}_2}} \right\}\) is the point in \(S\) at \(m = 1\).
\({{\mathop{\rm p}\nolimits} _3}\) is the point in \(S\) at \(m = 0\).

Step by step solution

The maximum and minimum is attained at an extreme point

Theorem 16states that consider f as a linear functional defined on a nonempty compact convex set \(S\).

Then, there are extreme points \(\widehat {\mathop{\rm v}\nolimits} \) and \(\widehat {\mathop{\rm w}\nolimits} \) of \(S\) such that \(f\left( {\widehat {\mathop{\rm v}\nolimits} } \right) = \mathop {\max }\limits_{{\mathop{\rm v}\nolimits} \in S} f\left( {\mathop{\rm v}\nolimits} \right)\) and \(f\left( {\widehat {\mathop{\rm w}\nolimits} } \right) = \mathop {\min }\limits_{{\mathop{\rm v}\nolimits} \in S} f\left( {\mathop{\rm v}\nolimits} \right)\).

Determine the maximum value \(m\) of \(f\)

According to theorem 16, the maximum value is attained at one of the extreme points of \(S\).

Evaluate \(f\) at the extreme point and select the largest value to find \(m\) as shown below:

\({f_1}\left( {{{\mathop{\rm p}\nolimits} _1}} \right) = - 1\), \({f_1}\left( {{{\mathop{\rm p}\nolimits} _2}} \right) = 3\), \({f_1}\left( {{{\mathop{\rm p}\nolimits} _3}} \right) = 3\), therefore, \({m_1} = 3\). Graph the line \({f_1}\left( {{x_1},{x_2}} \right) = {m_1}\) which means that \({x_1} + {x_2} = 3\), and \({\mathop{\rm x}\nolimits} = {{\mathop{\rm p}\nolimits} _2}\) is the only point in \(S\) at which \({f_1}\left( {\mathop{\rm x}\nolimits} \right) = 3\).

b. \({f_2}\left( {{{\mathop{\rm p}\nolimits} _1}} \right) = 1\), \({f_2}\left( {{{\mathop{\rm p}\nolimits} _2}} \right) = 1\), \({f_2}\left( {{{\mathop{\rm p}\nolimits} _3}} \right) = - 1\), therefore, \({m_2} = 1\). Graph the line \({f_2}\left( {{x_1},{x_2}} \right) = {m_2}\) which means that \({x_1} - {x_2} = 1\), and \({\mathop{\rm x}\nolimits} = {{\mathop{\rm p}\nolimits} _1}\) is the only point in \(S\) at which \({f_2}\left( {\mathop{\rm x}\nolimits} \right) = 1\).

c. \({f_3}\left( {{{\mathop{\rm p}\nolimits} _1}} \right) = - 1\), \({f_3}\left( {{{\mathop{\rm p}\nolimits} _2}} \right) = - 3\), \({f_3}\left( {{{\mathop{\rm p}\nolimits} _3}} \right) = 0\), therefore, \({m_3} = 0\). Graph the line \({f_3}\left( {{x_1},{x_2}} \right) = {m_3}\) which means that \( - 2{x_1} + {x_2} = 0\), and \({\mathop{\rm x}\nolimits} = {{\mathop{\rm p}\nolimits} _3}\) is the only point in \(S\) at which \({f_3}\left( {\mathop{\rm x}\nolimits} \right) = 0\).