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
Transcribed Image Text: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}.
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}
that is such that S CE2.
4. Derive a polyhedron P3 C R2 that is such that S C P3 and that is the smallest possible.
5. Represent the sets S, E1, E2 and P3 graphically.
6. Consider the three optimization problems
=
: min{(x − 3)² + (y - 2)² |(x, y) = E₁}
: min{(x − 3)² + (y − 2)² | (x, y) = E₂}
: min{(x − 3)² + (y - 2)² |(x, y) = P3}.
2₂ :=
Argue that z z3 ≥ zi and z* ≥ 23 ≥ 22.
7. The set S can be decomposed into S₁ and S₂. Obtain a lower bound on z* by solving two convex
optimization problems: one having S₁ as feasible region and the other having S2 as feasible region.
8. Use the above discussion to obtain the value of z*.
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