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Chapter 8: Q3E (page 437)

Question 3: Repeat Exercise 1 where \(m\) is the minimum value of f on \(S\) instead of the maximum value.

Short Answer

Expert verified
  1. \({{\mathop{\rm p}\nolimits} _3}\) is the point in \(S\) at \(m = - 3\).
  2. \(m = 1\)is the minimum value on the set \({\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _3}} \right\}\).
  3. \(m = - 3\) is the minimum value on the set \({\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2}} \right\}\).

Step by step solution

01

Repeat the given as in Exercise 1

Given points \({{\mathop{\rm p}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right),{\rm{ }}{{\mathop{\rm p}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\3\end{array}} \right),\) and \({{\mathop{\rm p}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right)\) in \({\mathbb{R}^2}\), 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. \(f\left( {{x_1},{x_2}} \right) = - 3{x_1} + {x_2}\)

02

The maximum and minimum is attained at an extreme point

Theorem 16states that let \(f\) be a linear functionaldefined 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 \(f\left( {\widehat {\mathop{\rm w}\nolimits} } \right) = \mathop {\min }\limits_{{\mathop{\rm v}\nolimits} \in S} f\left( {\mathop{\rm v}\nolimits} \right)\).

03

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

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

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

a. \({f_1}\left( {{{\mathop{\rm p}\nolimits} _1}} \right) = 1\), \({f_1}\left( {{{\mathop{\rm p}\nolimits} _2}} \right) = - 1\), \({f_1}\left( {{{\mathop{\rm p}\nolimits} _3}} \right) = - 3\), 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 \({\mathop{\rm x}\nolimits} = {{\mathop{\rm p}\nolimits} _3}\) is the only point in \(S\) at which \({f_1}\left( {\mathop{\rm x}\nolimits} \right) = - 3\).

b. \({f_2}\left( {{{\mathop{\rm p}\nolimits} _1}} \right) = 1\), \({f_2}\left( {{{\mathop{\rm p}\nolimits} _2}} \right) = 5\), \({f_2}\left( {{{\mathop{\rm p}\nolimits} _3}} \right) = 1\), 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 \({\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _3}} \right\}\) is the set in \(S\) at which \({f_2}\left( {\mathop{\rm x}\nolimits} \right) = 1\).


c. \({f_3}\left( {{{\mathop{\rm p}\nolimits} _1}} \right) = - 3\), \({f_3}\left( {{{\mathop{\rm p}\nolimits} _2}} \right) = - 3\), \({f_3}\left( {{{\mathop{\rm p}\nolimits} _3}} \right) = 5\), therefore, \({m_3} = - 3\). Graph the line \({f_3}\left( {{x_1},{x_2}} \right) = {m_3}\) which means that \( - 3{x_1} + {x_2} = - 3\), and \({\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2}} \right\}\) is the set in \(S\) at which \({f_3}\left( {\mathop{\rm x}\nolimits} \right) = - 3\).


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Most popular questions from this chapter

In Exercises 1-6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points.

5.\(\left( {\begin{aligned}{{}}1\\0\\{ - 2}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\1\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 1}\\5\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\5\\{ - 3}\end{aligned}} \right)\)

Question: 17. Choose a set \(S\) of three points such that aff \(S\) is the plane in \({\mathbb{R}^3}\) whose equation is \({x_3} = 5\). Justify your work.

Question: 30. Prove that the convex hull of a bounded set is bounded.

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d, so that \(\left( {f:d} \right)\) the hyperplane H described in the exercise.

Let A be the \({\bf{1}} \times {\bf{5}}\) matrix \(\left( {\begin{array}{*{20}{c}}{\bf{2}}&{\bf{5}}&{ - {\bf{3}}}&{\bf{0}}&{\bf{6}}\end{array}} \right)\). Note that \({\bf{Nul}}\,\,A\) is in \({\mathbb{R}^{\bf{5}}}\). Let \(H = {\bf{Nul}}\,\,A\).

Question: Let \({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{3}}}\\{\bf{1}}\\{\bf{2}}\end{array}} \right)\), \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{ - {\bf{1}}}\\{\bf{3}}\end{array}} \right)\), \({{\bf{n}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{\bf{4}}\\{\bf{2}}\end{array}} \right)\), and \({{\bf{n}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{3}}\\{\bf{1}}\\{\bf{5}}\end{array}} \right)\), let \({H_{\bf{1}}}\) be the hyperplane in \({\mathbb{R}^{\bf{4}}}\) through \({{\bf{p}}_{\bf{1}}}\) with normal \({{\bf{n}}_{\bf{1}}}\), and let \({H_{\bf{2}}}\) be the hyperplane through \({{\bf{p}}_{\bf{2}}}\) with normal \({{\bf{n}}_{\bf{2}}}\). Give an explicit description of \({H_{\bf{1}}} \cap {H_{\bf{2}}}\). (Hint: Find a point p in \({H_{\bf{1}}} \cap {H_{\bf{2}}}\) and two linearly independent vectors \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) that span a subspace parallel to the 2-dimensional flat \({H_{\bf{1}}} \cap {H_{\bf{2}}}\).)
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