System of Linear Inequalities q Linear Programming 2012

System of Linear Inequalities q Linear Programming 2012 1

q Questions: Ø Are there any other representations describing the same set? Ø How can we identify the different representation given a representation of a set? Ø Which are the most important elements in a representation to describe the set and which elements are redundant or unnecessary? Ø Given an instance of a representation, does it have a feasible solution or not? Ø How can we verify that it has a feasible solution or not? Linear Programming 2012 2

q References: Convexity and Optimization in Finite Dimensions 1, Josef Stoer and Christoph Witzgall, 1970, Springer-Verlag. Convex Analysis, R. Tyrrell Rockafellar, 1970, Princeton University Press. Integer and Combinatorial Optimization, George L. Nemhauser, Laurence A. Wolsey, 1988, Wiley. Theory of Linear and Integer Programming, Alexander Schrijver, 1986, Wiley. Linear Programming 2012 3

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Convex Sets q Linear Programming 2012 7

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Subspaces q Linear Programming 2012 9

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q Picture: Linear Programming 2012 16

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Review q Linear Programming 2012 18

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Back to subspace q Linear Programming 2012 21

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Remarks q Linear Programming 2012 25

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Obtaining constrained form from generators (1, -3, 1) Linear Programming 2012 0 27

Remarks q Linear Programming 2012 28