Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Please answer question 4
![Consider the set \( S \) composed of the points on the line segment \( S_1 \) between points \((-1,0)\) and \((2,0)\) and of the points on the line segment \( S_2 \) between points \((0,-1)\) and \((0,3)\). We are interested in the optimization problem
\[
z^* := \min \{ (x - 3)^2 + x^2 y^2 + (y - 2)^2 \mid (x, y) \in S \}.
\]
Note that \( z^* \leq 5 \) since \((x, y) = (2, 0)\) is a feasible solution to this problem.
1. Determine conditions on the values of \( a, b, c \) and \( d \) for which the set
\[
E = \{ (x, y) \in \mathbb{R}^2 \mid a(x-c)^2 + b(y-d)^2 \leq 1 \}
\]
is convex and is such that \( S \subseteq E \).
2. Determine the values of \( a \) and \( b \) that lead to the smallest (based on area) convex set
\[
E_1 = \{ (x, y) \in \mathbb{R}^2 \mid a(x)^2 + b(y)^2 \leq 1 \}
\]
that is such that \( S \subseteq E_1 \).
3. Determine the values of \( a \) and \( b \) that lead to the smallest (based on area) convex set
\[
E_2 = \{ (x, y) \in \mathbb{R}^2 \mid a(x-1)^2 + b(y-1)^2 \leq 1 \}
\]
that is such that \( S \subseteq E_2 \).
4. Derive a polyhedron \( P_3 \subseteq \mathbb{R}^2 \) that is such that \( S \subseteq P_3 \) and that is the smallest possible.
5. Represent the sets \( S, E_1, E_2, \) and \( P_3 \) graphically](https://content.bartleby.com/qna-images/question/fe12c986-f40c-4f0d-b28b-0f5656c9be52/61e799c1-12ac-4804-8863-ced3a4b50bdd/8xfuz_processed.jpeg)
Transcribed Image Text:Consider the set \( S \) composed of the points on the line segment \( S_1 \) between points \((-1,0)\) and \((2,0)\) and of the points on the line segment \( S_2 \) between points \((0,-1)\) and \((0,3)\). We are interested in the optimization problem
\[
z^* := \min \{ (x - 3)^2 + x^2 y^2 + (y - 2)^2 \mid (x, y) \in S \}.
\]
Note that \( z^* \leq 5 \) since \((x, y) = (2, 0)\) is a feasible solution to this problem.
1. Determine conditions on the values of \( a, b, c \) and \( d \) for which the set
\[
E = \{ (x, y) \in \mathbb{R}^2 \mid a(x-c)^2 + b(y-d)^2 \leq 1 \}
\]
is convex and is such that \( S \subseteq E \).
2. Determine the values of \( a \) and \( b \) that lead to the smallest (based on area) convex set
\[
E_1 = \{ (x, y) \in \mathbb{R}^2 \mid a(x)^2 + b(y)^2 \leq 1 \}
\]
that is such that \( S \subseteq E_1 \).
3. Determine the values of \( a \) and \( b \) that lead to the smallest (based on area) convex set
\[
E_2 = \{ (x, y) \in \mathbb{R}^2 \mid a(x-1)^2 + b(y-1)^2 \leq 1 \}
\]
that is such that \( S \subseteq E_2 \).
4. Derive a polyhedron \( P_3 \subseteq \mathbb{R}^2 \) that is such that \( S \subseteq P_3 \) and that is the smallest possible.
5. Represent the sets \( S, E_1, E_2, \) and \( P_3 \) graphically
Expert Solution
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Step 1: Convex hull
The The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space,
or equivalently, as the set of all convex combinations of points in the subset.
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