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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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Maxima, Minima, Suprema, In°ma 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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Maxima, Minima, Suprema, In°ma However, Maxima and Minima may not be de°ned for some sets. For 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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Maxima, Minima, Suprema, In°ma 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 3 , ³ 2 , ³ 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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Maxima, Minima, Suprema, In°ma De°ne 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 3 , ³ 2 , ³ 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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Maxima, Minima, Suprema, In°ma the lower bounds of A : L . A / D f a 2 R : a ² 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 . ³1 , 0 ] I B D . 0 , 1 / and L . B / D . ³1 , 0 ] I Z Df ..., ³ 3 , ³ 2 , ³ 1 , 0 , 1 , 2 , 3 ,... g and L . Z / D ? I Xiong (Rice University) ECON 401 August 31, 2011 6 / 22
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Maxima, Minima, Suprema, In°ma De°ne inf A D ° max L . A / if L . A / 6D ? , ³1 if L . A / D ? .
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