Lecture05 - IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis...

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IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis Kara Lecture # 5 Date: 7 October 2007 1

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HYPERPLANES and HALFSPACES Let be given The set is called a hyperplane. A hyperplane divides R n into two regions: called halfspaces . Hence a halfspace is a collection of the points of the form } ˆ : { b x a R x T n } ˆ : { b x a R x T n 2 R b R a n ˆ and
POLYHEDRON Let A Polyhedron or a polyhedral set is a set that can be described in the form: i.e. intersection of a finite number of halfspaces. } : { b Ax R x n 1 1 ˆ b x a T 3 3 ˆ b x a T 2 2 ˆ b x a T 3 m mXn R b R A and

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POLYTOPE A set S in R n is bounded if there exists a constant K such that the absolute value of every element of S is less than or equal to K. A polytope is a bounded poyhedron. 4
CONVEXITY Any point of the form for λЄ [0 1] is called a convex combination of x and y. If λЄ (0 1) then the convex combination is called strict . y x ) 1 ( 5

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CONVEXITY Let be given s.t The vector is said to be a convex combination of The convex hull of is the set of all convex combinations of these vectors. 6 k n k R R x x x and ,..., , 2 1 1 1 T k i i i x 1 k x x x ,..., , 2 1 k x x x ,..., , 2 1
CONVEXITY A set S in R n is called a convex set if for any x,y Є S then Note that for λЄ [0 1] represents a point on the line segment joining x and y. 1] [0 each for , ) 1 (   S y x y x ) 1 ( 7

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CONVEXITY Thus, a set is convex if the segment joining any two of its elements is contained in the set. Convex Not Convex 8
Properties of Convex sets The following properties can be derived from the definitions. a)

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Lecture05 - IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis...

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