# hw1 - (e An unbounded n-dimensional polyhedral set can have...

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HOMEWORK 1 Due February 8 For students registered for 3 credits: 1.3, 1.10, 1.11, 1.15, 2.1, 2.9, (1), (2), and (3). For students registered for 4 credits: 1.5, 1.7, 1.10, 1.11, 1.12, 1.15, 2.1, 2.6, 2.9, (1), (2) and (3). (1) Find all extreme points and extreme directions of the following polyhedral set: X = { ( x 1 ,x 2 ,x 3 ,x 4 ) : x 1 ,x 2 ,x 3 ,x 4 0 , - x 1 + x 2 - 2 x 3 1 , - 2 x 1 - x 3 + 2 x 4 2 } (2) Answer the following questions and provide a brief explanation or illustra- tion. (a) Is it possible for X = { x : Ax b,x 0 } to be empty but D = { d : Ad 0 , 1 d = 1 ,d 0 } to be nonempty? (b) Is there a relationship between redundancy and degeneracy of a poly- hedral set? (c) Does degeneracy imply redundancy in two dimensions? (d) If the intersection of a ﬁnite number of halfspaces is nonempty, then this set has at least one extreme point. True or false?
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Unformatted text preview: (e) An unbounded n-dimensional polyhedral set can have at most n ex-treme directions. True or false? (f) What is the maximum (actual) dimension of X = { x : Ax = b,x ≥ } , where A is m × n of rank t , t ≤ m ≤ n ? (3) Consider the following table production problem. Table one can be sold at 10 per unit, which requires 2 units of wood one and 1 unit of wood two. Similarly, Table one can be sold at 14 per unit, which requires 1 units of wood one and 2 unit of wood two. There are three agents each of which controls a resource vector (2 , 4), (4 , 2) and (4 , 4) respectively. Describe the core of the resulting cooperative game. 1...
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