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slide5-aug31 - ECON 401 Optimization problems Siyang Xiong...

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ECON 401 Optimization problems Siyang Xiong Rice University August 31, 2011 Xiong (Rice University) ECON 401 August 31, 2011 1 / 22
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Given a nonempty A R , max A D f a 2 A : a ± b for any b 2 A g ; min A D f a 2 A : a ² b for any b 2 A g ; for example, A D [ 0 , 1 ] , then max A D 1 I min A D 0 . Xiong (Rice University) ECON 401 August 31, 2011 2 / 22
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example, B D . 0 , 1 / max B and min B do not exist. Xiong (Rice University) ECON 401 August 31, 2011 3 / 22
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the upper bounds of A : U . A / D f a 2 R : a b for any b 2 A g . if U . A / 6D ? , we say A is bounded above. For example: A D [ 0 , 1 ] and U . A / D [ 1 , C1 / I B D . 0 , 1 / and U . B / D [ 1 , C1 / I Z Df ..., ± 3 , ± 2 , ± 1 , 0 , 1 , 2 , 3 , ... g and U . Z / D ? I Xiong (Rice University) ECON 401 August 31, 2011 4 / 22
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sup A D min U . A / if U . A / 6D ? , C1 if U . A / D ? . For example: A D [ 0 , 1 ] and U . A / D [ 1 , C1 / , hence sup A D 1; B D . 0 , 1 / and U . B / D [ 1 , C1 / , hence sup B D 1; Z Df ..., 3 , 2 , 1 , 0 , 1 , 2 , 3 , ... g and U . Z / D ? , hence sup Z D C1 . Xiong (Rice University) ECON 401 August 31, 2011 5 / 22
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the lower bounds of A : L . A / D f a 2 R : a b for any b 2 A g . if L . A / 6D ? , we say A is bounded below. For example: A D [ 0 , 1 ] and L . A / D . ±1 , 0 ] I B D . 0 , 1 / and L . B
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This note was uploaded on 02/09/2012 for the course ECON 401 taught by Professor Siyang during the Spring '11 term at Rice.

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slide5-aug31 - ECON 401 Optimization problems Siyang Xiong...

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