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
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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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