ConvexSet - ESI 6417 Convex Sets: - Convex and Polyhedral...

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ESI 6417 Convex and Polyhedral Sets 1 Ref. Chapter 2, BJS Convex Sets: - A set X in R n is called a “convex” set if, for any (or every) two points (vectors) x 1 and x 2 in X , any convex combination of x 1 and x 2 lies in X . ± A convex combination = λ x 1 + (1 – ) x 2 , where [0, 1]. ± A convex combination of x 1 and x 2 represents a point on the line segment joining x 1 and x 2 . ± Which of the following sets are convex? ± If X and Y are two convex sets, is X Y convex? ± If X and Y are two convex sets, is X Y convex? - Extreme points of a convex set: A point x X is called an “extreme” point of X if x cannot be represented as a convex combination of two distinct points in X . ± Identify extreme points of the following sets. Hyperplanes and halfspaces - A set of the form { x R n : a T x = α }, where a is a vector and is a constant, is called a “hyperplane.” For example, ± X ={ x R 2 : 2 x 1 + 1 x 2 = 2}
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ESI 6417 Convex and Polyhedral Sets
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This note was uploaded on 09/19/2008 for the course ESI 6417 taught by Professor Siriphonglawphongpanich during the Spring '07 term at University of Florida.

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ConvexSet - ESI 6417 Convex Sets: - Convex and Polyhedral...

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