Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Chapter 27, Problem 6A
To determine
To multiply
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Consider the cones
K =
= {(x1, x2, x3) | € R³ :
X3
≥√√√2x² + 3x²
M =
= {(21,22,23)
(x1, x2, x3) Є R³: x3 >
+
2
3
Prove that M = K*.
Hint: Adapt the proof from the lecture notes for finding the dual of the Lorentz cone. Alternatively, prove the
formula (AL)* = (AT)-¹L*, for any cone LC R³ and any 3 × 3 nonsingular matrix A with real entries, where
AL = {Ax = R³ : x € L}, and apply it to the 3-dimensional Lorentz cone with an appropriately chosen matrix
A.
I am unable to solve part b.
Let
M = M₁U M₂ UM3 and K
M₁ = {(x1, x2) ER²: 2 ≤ x ≤ 8, 2≤ x ≤8},
M₂ = {(x1, x2)™ € R² : 4 ≤ x₁ ≤ 6, 0 ≤ x2 ≤ 10},
M3 = {(x1, x2) Є R²: 0 ≤ x₁ ≤ 10, 4≤ x ≤ 6},
¯ = cone {(1, 2), (1,3)†} ≤ R².
(a) Determine the set E(M,K) of efficient points of M with respect to K.
(b) Determine the set P(M, K) of properly efficient points of M with respect to K.
Chapter 27 Solutions
Mathematics For Machine Technology
Ch. 27 - Prob. 1ACh. 27 - Of the 2478 pieces manufactured one day, the...Ch. 27 - Prob. 3ACh. 27 - Prob. 4ACh. 27 - Prob. 5ACh. 27 - Prob. 6ACh. 27 - In Exercises 7 through 12, express each of the...Ch. 27 - Prob. 8ACh. 27 - In Exercises 7 through 12, express each of the...Ch. 27 - In Exercises 7 through 12, express each of the...
Ch. 27 - In Exercises 7 through 12, express each of the...Ch. 27 - Prob. 12ACh. 27 - Prob. 13ACh. 27 - In Exercises 13 through 22, perform the indicated...Ch. 27 - Prob. 15ACh. 27 - Prob. 16ACh. 27 - In Exercises 13 through 22, perform the indicated...Ch. 27 - Prob. 18ACh. 27 - Prob. 19ACh. 27 - Prob. 20ACh. 27 - Prob. 21ACh. 27 - In Exercises 13 through 22, perform the indicated...Ch. 27 - An aluminum slab 0.082 meter thick is machined...Ch. 27 - A piece of sheet metal is 1.12 meters wide. Strips...Ch. 27 - In Exercises 25 through 28, express each of the...Ch. 27 - In Exercises 25 through 28, express each of the...Ch. 27 - In Exercises 25 through 28, express each of the...Ch. 27 - Prob. 28ACh. 27 - In Exercises 29 through 32, express each of the...Ch. 27 - In Exercises 29 through 32, express each of the...Ch. 27 - In Exercises 29 through 32, express each of the...Ch. 27 - Prob. 32ACh. 27 - Determine the total length of stock in inches...Ch. 27 - Prob. 34ACh. 27 - Prob. 35A
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