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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

Let\({v_1} = \left[ {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{array}} \right]\),\({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{0}}\end{array}} \right]\), and let\(S = \left\{ {{v_1},{v_2},{v_3}} \right\}\).

  1. Show that the set is affinely independent.
  2. Find the barycentric coordinates of\({p_1} = \left[ {\begin{array}{*{20}{c}}2\\3\end{array}} \right]\),\({p_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{array}} \right]\),\({p_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{\bf{1}}\end{array}} \right]\),\({p_{\bf{4}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{1}}}\end{array}} \right]\), and\({p_{\bf{5}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{1}}\end{array}} \right]\), with respect to S.
  3. Let\(T\)be the triangle with vertices\({v_1}\),\({v_{\bf{2}}}\), and\({v_{\bf{3}}}\). When the sides of\(T\)are extended, the lines divide\({\mathbb{R}^{\bf{2}}}\)into seven regions. See Figure 8. Note the signs of the barycentric coordinates of the points in each region. For example,\({{\bf{p}}_{\bf{5}}}\)is inside the triangle\(T\)and all its barycentric coordinates are positive. Point\({{\bf{p}}_{\bf{1}}}\)has coordinates\(\left( { - , + , + } \right)\). Its third coordinate is positive because\({{\bf{p}}_{\bf{1}}}\)is on the\({{\bf{v}}_{\bf{3}}}\)side of the line through\({{\bf{v}}_{\bf{1}}}\)and\({{\bf{v}}_{\bf{2}}}\). Its first coordinate is negative because\({{\bf{p}}_{\bf{1}}}\)is opposite the\({{\bf{v}}_{\bf{1}}}\)side of the line through\({{\bf{v}}_{\bf{2}}}\)and\({{\bf{v}}_{\bf{3}}}\). Point\({{\bf{p}}_{\bf{2}}}\)is on the\({{\bf{v}}_{\bf{2}}}{{\bf{v}}_{\bf{3}}}\)edge of\(T\). Its coordinates are\(\left( {0, + , + } \right)\). Without calculating the actual values, determine the signs of the barycentric coordinates of points\({{\bf{p}}_{\bf{6}}}\),\({{\bf{p}}_{\bf{7}}}\), and\({{\bf{p}}_{\bf{8}}}\)as shown in Figure 8.

Question: 29. Prove that the open ball \(B\left( {{\rm{p}},\delta } \right) = \left\{ {{\rm{x:}}\left\| {{\rm{x - p}}} \right\| < \delta } \right\}\)is a convex set. (Hint: Use the Triangle Inequality).

In Exercises 1-4, write y as an affine combination of the other point listed, if possible.

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{2}}}\\{\bf{2}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{4}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{7}}\end{aligned}} \right)\), \({\bf{y}} = \left( {\begin{aligned}{*{20}{c}}{\bf{5}}\\{\bf{3}}\end{aligned}} \right)\)

The parametric vector form of a B-spline curve was defined in the Practice Problems as

\({\bf{x}}\left( t \right) = \frac{1}{6}\left[ \begin{array}{l}{\left( {1 - t} \right)^3}{{\bf{p}}_o} + \left( {3t{{\left( {1 - t} \right)}^2} - 3t + 4} \right){{\bf{p}}_1}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {3{t^2}\left( {1 - t} \right) + 3t + 1} \right){{\bf{p}}_2} + {t^3}{{\bf{p}}_3}\end{array} \right]\;\), for \(0 \le t \le 1\) where \({{\bf{p}}_o}\) , \({{\bf{p}}_1}\), \({{\bf{p}}_2}\) , and \({{\bf{p}}_3}\) are the control points.

a. Show that for \(0 \le t \le 1\), \({\bf{x}}\left( t \right)\) is in the convex hull of the control points.

b. Suppose that a B-spline curve \({\bf{x}}\left( t \right)\)is translated to \({\bf{x}}\left( t \right) + {\bf{b}}\) (as in Exercise 1). Show that this new curve is again a B-spline.

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