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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

In Exercises 9 and 10, mark each statement True or False. Justify each answer.

9.

a. If \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}\) are in \({\mathbb{R}^n}\) and if the set \(\left\{ {{{\mathop{\rm v}\nolimits} _1} - {{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _2},...,{{\mathop{\rm v}\nolimits} _p} - {{\mathop{\rm v}\nolimits} _2}} \right\}\) is linearly dependent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is affinely dependent. (Read this carefully.)

b. If \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}\) are in \({\mathbb{R}^n}\) and if the set of homogeneous forms \(\left\{ {{{\overline {\mathop{\rm v}\nolimits} }_1},...,{{\overline {\mathop{\rm v}\nolimits} }_p}} \right\}\) in \({\mathbb{R}^{n + 1}}\) is linearly independent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is affinely dependent.

c. A finite set of points \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _k}} \right\}\) is affinely dependent if there exist real numbers \({c_1},...,{c_k}\) , not all zero, such that \({c_1} + ... + {c_k} = 1\) and \({c_1}{{\mathop{\rm v}\nolimits} _1} + ... + {c_k}{{\mathop{\rm v}\nolimits} _k} = 0\).

d. If \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is an affinely independent set in \({\mathbb{R}^n}\) and if p in \({\mathbb{R}^n}\) has a negative barycentric coordinate determined by S, then p is not in \({\mathop{\rm aff}\nolimits} S\).

e.

If \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},a,\) and \(b\) are in \({\mathbb{R}^3}\) and if ray \({\mathop{\rm a}\nolimits} + t{\mathop{\rm b}\nolimits} \) for \(t \ge 0\) intersects the triangle with vertices \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},\) and \({{\mathop{\rm v}\nolimits} _3}\) then the barycentric coordinates of the intersection points are all nonnegative.

Let\(\left\{ {{p_1},{p_2},{p_3}} \right\}\)be an affinely dependent set of points in\({\mathbb{R}^{\bf{n}}}\)and let\(f:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\)be a linear transformation. Show that\(\left\{ {f\left( {{{\bf{p}}_1}} \right),f\left( {{{\bf{p}}_2}} \right),f\left( {{{\bf{p}}_3}} \right)} \right\}\)is affinely dependent in\({\mathbb{R}^{\bf{m}}}\).

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

In Exercises 21-26, prove the given statement about subsets A and B of \({\mathbb{R}^n}\), or provide the required example in \({\mathbb{R}^2}\). A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).

25. \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\)

Question: 20. Let \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\) is a linear transformation, and let \(T\) be an affine subset of \({\mathbb{R}^{\bf{m}}}\), and let \(S = \left\{ {{\bf{x}} \in {\mathbb{R}^n}\,:\,f\left( {\bf{x}} \right) \in T} \right\}\). Show that \(S\) is an affine subset of \({\mathbb{R}^m}\).
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