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Chapter 8: Q7E (page 437)

Question: In Exercises 5-8, find the minimal representation of the polytope defined by the inequalities \(A{\mathop{\rm x}\nolimits} \le {\mathop{\rm b}\nolimits} \) and \({\mathop{\rm x}\nolimits} \ge 0\).

7. \(A = \left( {\begin{array}{*{20}{c}}1&3\\1&1\\4&1\end{array}} \right),{\rm{ }}{\mathop{\rm b}\nolimits} = \left( {\begin{array}{*{20}{c}}{18}\\{10}\\{28}\end{array}} \right)\)

Short Answer

Expert verified

The minimal representation of the polytope \(P\) is \(\left\{ {\left( {\begin{array}{*{20}{c}}0\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}7\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}6\\4\end{array}} \right),\left( {\begin{array}{*{20}{c}}0\\6\end{array}} \right)} \right\}\).

Step by step solution

01

The three inequalities in \(A{\mathop{\rm x}\nolimits}  \le {\mathop{\rm b}\nolimits} \)

The three matrix inequalities \(A{\mathop{\rm x}\nolimits} \le {\mathop{\rm b}\nolimits} \) yield the following system of inequalities:

  1. \({x_1} + 3{x_2} \le 18\)
  2. \({x_1} + {x_2} \le 10\)
  3. \(4{x_1} + {x_2} \le 28\)
02

Determine the \({{\mathop{\rm x}\nolimits} _1}\)-intercept and \({{\mathop{\rm x}\nolimits} _2}\)-intercept of the three lines

The condition \({\mathop{\rm x}\nolimits} \ge 0\) places polytope \(P\) in the first quadrant of the plane. One vertex is \(\left( {0,0} \right)\).

The \({{\mathop{\rm x}\nolimits} _1}\)-intercepts\(\left( {{\mathop{\rm If}\nolimits} {{\mathop{\rm x}\nolimits} _2} = 0} \right)\) of the three lines are 18, 10, and 7, so \(\left( {7,0} \right)\) is a vertex. The \({{\mathop{\rm x}\nolimits} _2}\)-intercepts \(\left( {{\mathop{\rm If}\nolimits} {{\mathop{\rm x}\nolimits} _1} = 0} \right)\) of the three lines are 6, 10, and 28, then \(\left( {0,6} \right)\) is a vertex.

03

Determine the intersection point corresponds to inequalities

The intersection of (a) and (b) is at \({{\mathop{\rm P}\nolimits} _{{\mathop{\rm a}\nolimits} {\mathop{\rm b}\nolimits} }} = \left( {6,4} \right)\). Testing \({{\mathop{\rm P}\nolimits} _{{\mathop{\rm ab}\nolimits} }}\) in (c) gives \(4\left( 6 \right) + 4 = 28\), so \({{\mathop{\rm P}\nolimits} _{{\mathop{\rm ab}\nolimits} }}\) is in \({\mathop{\rm P}\nolimits} \).

04

Determine the minimal representation of the polytope

The set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},..,{{\mathop{\rm v}\nolimits} _k}} \right\}\) is aminimal representation of the polytope\(P\)when\(P = {\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _k}} \right\}\) and for each \(i = 1,...,k\), \({{\mathop{\rm v}\nolimits} _1} \notin \left\{ {{{\mathop{\rm v}\nolimits} _j}:j \ne i} \right\}\).

The four vertices of the polytope are \(\left( {0,0} \right),\left( {7,0} \right)\left( {6,4} \right),\,\,{\mathop{\rm and}\nolimits} \,\,\left( {0,6} \right)\).

Thus, the minimal representation of the polytope \(P\) is \(\left\{ {\left( {\begin{array}{*{20}{c}}0\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}7\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}6\\4\end{array}} \right),\left( {\begin{array}{*{20}{c}}0\\6\end{array}} \right)} \right\}\).

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Most popular questions from this chapter

Question: Let \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{1}}\\{\bf{2}}\end{array}} \right)\), \({\bf{n}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{5}}\\{ - {\bf{1}}}\end{array}} \right)\), \({{\bf{v}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{1}}\\{\bf{1}}\\{\bf{1}}\end{array}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{0}}\\{\bf{1}}\\{\bf{3}}\end{array}} \right)\), and \({{\bf{v}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{4}}\\{\bf{0}}\\{\bf{4}}\end{array}} \right)\), and let H be the hyperplane in\({\mathbb{R}^{\bf{4}}}\) with normal n and passing through p. Which of the points \({{\bf{v}}_{\bf{1}}}\), \({{\bf{v}}_{\bf{2}}}\), and \({{\bf{v}}_{\bf{3}}}\) are on the same side of H as the origin, and which are not?

Questions: Let \({F_{\bf{1}}}\) and \({F_{\bf{2}}}\) be 4-dimensional flats in \({\mathbb{R}^{\bf{6}}}\), and suppose that \({F_{\bf{1}}} \cap {F_{\bf{2}}} \ne \phi \). What are the possible dimension of \({F_{\bf{1}}} \cap {F_{\bf{2}}}\)?

Question: In Exercises 21 and 22, mark each statement True or False. Justify each answer.

22 a. If \(d\)is a real number and \(f\) is a nonzero linear functional defined on \({\mathbb{R}^n}\) , then \(f:d\)is a hyperplane in \({\mathbb{R}^n}\) .

b. Given any vector n and any real number \(d\), the set \(\left\{ {x:n \cdot x = d} \right\}\) is a hyperplane.

c. If \(A\) and \(B\) are nonempty disjoint sets such that \(A\) is compact and \(B\) is closed, then there exists a hyperplane that strictly separates \(A\) and \(B\).

d. If there exists a hyperplane \(H\) such that \(H\) does not strictly separate two sets \(A\) and \(B\), then \(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) \ne \emptyset \) and \(B\).

Question: In Exercise 4, determine whether each set is open or closed or neither open nor closed.

4. a. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} = {\bf{1}}} \right\}\)

b. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} > {\bf{1}}} \right\}\)

c. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} \le {\bf{1}}\,\,\,and\,\,y > {\bf{0}}} \right\}\)

d. \(\left\{ {\left( {x,y} \right):y \ge {x^{\bf{2}}}} \right\}\)

e. \(\left\{ {\left( {x,y} \right):y < {x^{\bf{2}}}} \right\}\)

Question: Let \({{\bf{v}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{\bf{3}}\\{\bf{0}}\end{array}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{1}}}\\{\bf{0}}\\{\bf{4}}\end{array}} \right)\), and \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{2}}\\{\bf{1}}\\{\bf{1}}\end{array}} \right)\)

\({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{ - {\bf{3}}}\\{\bf{5}}\\{\bf{3}}\end{array}} \right)\) b. \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{9}}}\\{{\bf{10}}}\\{\bf{9}}\\{ - {\bf{13}}}\end{array}} \right)\) c. \({{\bf{p}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{2}}\\{\bf{8}}\\{\bf{5}}\end{array}} \right)\)

and \(S = \left\{ {{{\bf{v}}_1},\,\,{{\bf{v}}_2},\,{{\bf{v}}_3}} \right\}\). It can be shown that S is linearly independent.

a. Is \({{\bf{p}}_{\bf{1}}}\) is span S? Is \({{\bf{p}}_{\bf{1}}}\) is \({\bf{aff}}\,S\)?

b. Is \({{\bf{p}}_{\bf{2}}}\) is span S? Is \({{\bf{p}}_{\bf{2}}}\) is \({\bf{aff}}\,S\)?

c. Is \({{\bf{p}}_{\bf{3}}}\) is span S? Is \({{\bf{p}}_{\bf{3}}}\) is \({\bf{aff}}\,S\)?
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