Consider the set S composed of the points on the line segment S₁ between points (-1,0) and (2,0) and of the points on the line segment S2 between points (0, -1) and (0,3). We are interested in the optimization problem min{(x − 3)² + x²y² + (y-2)² | (x, y) = S}. E Note that z* 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) = R² | a(x - c)² + b(y - d)² ≤ 1} E is convex and is such that SCE. 2. Determine the values of a and b that lead to the smallest (based on area) convex set E₁ = {(x, y) = R² | a(x)² + b(y)² ≤ 1} that is such that SCE₁. 3. Determine the values of a and b that lead to the smallest (based on area) convex set E₂ = {(x, y) = R² | a(x - 1)² + b(y − 1)² ≤ 1} E that is such that S CE2. 4. Derive a polyhedron P3 CR2 that is such that S C P3 and that is the smallest possible.
Consider the set S composed of the points on the line segment S₁ between points (-1,0) and (2,0) and of the points on the line segment S2 between points (0, -1) and (0,3). We are interested in the optimization problem min{(x − 3)² + x²y² + (y-2)² | (x, y) = S}. E Note that z* 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) = R² | a(x - c)² + b(y - d)² ≤ 1} E is convex and is such that SCE. 2. Determine the values of a and b that lead to the smallest (based on area) convex set E₁ = {(x, y) = R² | a(x)² + b(y)² ≤ 1} that is such that SCE₁. 3. Determine the values of a and b that lead to the smallest (based on area) convex set E₂ = {(x, y) = R² | a(x - 1)² + b(y − 1)² ≤ 1} E that is such that S CE2. 4. Derive a polyhedron P3 CR2 that is such that S C P3 and that is the smallest possible.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe12c986-f40c-4f0d-b28b-0f5656c9be52%2F61e799c1-12ac-4804-8863-ced3a4b50bdd%2F8xfuz_processed.jpeg&w=3840&q=75)
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
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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