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- Some Geometry With Cones Problem Suppose that C is a cone in R" and let us define another set C' in R" as C = {ye R"ly¹x ≤ 0 for all x = C}. Prove that C' is also a cone in R". Determine the intersection between C and C'. Construct C' from C if C is the polyhedron cone XI C = - {[ + ]« R²|1 - 292 50,-25 + x2 50}. ≤0,-2x1 X2 and plot both C and C' in the R² plane.

- Some Geometry With Cones Problem Suppose that C is a cone in R" and let us define another set C' in R" as C = {ye R"ly¹x ≤ 0 for all x = C}. Prove that C' is also a cone in R". Determine the intersection between C and C'. Construct C' from C if C is the polyhedron cone XI C = - {[ + ]« R²|1 - 292 50,-25 + x2 50}. ≤0,-2x1 X2 and plot both C and C' in the R² plane.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...
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- Some Geometry With Cones Problem Suppose that C is a cone in R" and let us define another set C' in R" as C' = {y = R"y¹x ≤ 0 for all x € C}. Prove that C' is also a cone in R". Determine the intersection between C and C'. Construct C' from Cif C is the polyhedron cone XI C = - { [ ] = 4²| 41-2502 50,-241 + x2 50}. ≤0,-2x₁ X2 and plot both C and C' in the R² plane.
Transcribed Image Text:- Some Geometry With Cones Problem Suppose that C is a cone in R" and let us define another set C' in R" as C' = {y = R"y¹x ≤ 0 for all x € C}. Prove that C' is also a cone in R". Determine the intersection between C and C'. Construct C' from Cif C is the polyhedron cone XI C = - { [ ] = 4²| 41-2502 50,-241 + x2 50}. ≤0,-2x₁ X2 and plot both C and C' in the R² plane.
Cones A subset set C≤R" is a called a cone in R" if, for all x in C, 2x is also in C for all λ > 0. It should be noted that as long that C is not empty, the origin x = 0 must be in the cone. To see this, suppose that C is not empty so that some point x is in C, then 2x = C for all > > 0 and setting λ = 0, we find that Ox C or simply 0 C. Thus we find that C + Ø 0 € C. Of course 0 € C ⇒ C + Ø and so we see that C + Ø 0 € C. A cone that is also a convex set is called a convex cone and we say that a convex cone is pointed if it contains the origin 0 as an extreme point. A Polyhedral Cone A polyhedron of the form P = {x € R" | Ax ≥ 0} is a cone known as a polyhedral cone because if x = P, so that Ax ≥ 0, then (for all λ ≥ 0), 2Ax ≥ 20, or A(2x) ≥ 0, which says that λx € P. We note that P is not empty since x = 0 € P. Note that X1 X1 P = - {*]*[ [ * ] } X2 X2 is a polyhedron cone that is not pointed, and is shown in the figure below. ER² 1 1 P = P is everything above the line x₁ + x₂ = 0 and is not pointed Suppose also that A is a full rank m x n matrix with m≥n. Then since all the m constraints Ax ≥ 0 are active at x = 0 € R" and m≥n, we see that x = 0 will be an extreme point of P showing then that P is pointed for m≥n. For example, the cone 1 -{][I][:] {* R² X2 21 is a polyhedron cone that is pointed, and is shown in the figure below. XI X2 P is everything above the lines x₁ + x₂ = 0 and 2x1 + x₂ = 0 and is pointed 0 0 Theorem #10: Cones and Convex Sets A cone is a convex set if and only if it is closed under vector addition. Recall that a set in R¹ is closed under vector addition if for any xe C and y e C, x+y e C.
Transcribed Image Text:Cones A subset set C≤R" is a called a cone in R" if, for all x in C, 2x is also in C for all λ > 0. It should be noted that as long that C is not empty, the origin x = 0 must be in the cone. To see this, suppose that C is not empty so that some point x is in C, then 2x = C for all > > 0 and setting λ = 0, we find that Ox C or simply 0 C. Thus we find that C + Ø 0 € C. Of course 0 € C ⇒ C + Ø and so we see that C + Ø 0 € C. A cone that is also a convex set is called a convex cone and we say that a convex cone is pointed if it contains the origin 0 as an extreme point. A Polyhedral Cone A polyhedron of the form P = {x € R" | Ax ≥ 0} is a cone known as a polyhedral cone because if x = P, so that Ax ≥ 0, then (for all λ ≥ 0), 2Ax ≥ 20, or A(2x) ≥ 0, which says that λx € P. We note that P is not empty since x = 0 € P. Note that X1 X1 P = - {*]*[ [ * ] } X2 X2 is a polyhedron cone that is not pointed, and is shown in the figure below. ER² 1 1 P = P is everything above the line x₁ + x₂ = 0 and is not pointed Suppose also that A is a full rank m x n matrix with m≥n. Then since all the m constraints Ax ≥ 0 are active at x = 0 € R" and m≥n, we see that x = 0 will be an extreme point of P showing then that P is pointed for m≥n. For example, the cone 1 -{][I][:] {* R² X2 21 is a polyhedron cone that is pointed, and is shown in the figure below. XI X2 P is everything above the lines x₁ + x₂ = 0 and 2x1 + x₂ = 0 and is pointed 0 0 Theorem #10: Cones and Convex Sets A cone is a convex set if and only if it is closed under vector addition. Recall that a set in R¹ is closed under vector addition if for any xe C and y e C, x+y e C.
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