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Convexity basics

Convexity basics - Convex Sets and Functions Given the...

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Convex Sets and Functions Given the vectors x 1 , . . . , x m in Euclidean space R n and real numbers λ i 0 with m i =1 λ i = 1 , the vector sum λ 1 x 1 + . . . + λ m x m is called a convex combination of these points. For example, the convex combination of two points is the line segment between these two points, and the convex combination of three non–colinear points is a triangle. ESI 6492. Global Optimization Convex Sets and Functions – p. 1
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Convex Sets and Functions A subset C of R n is said to be convex if for every x 1 , x 2 C and λ R , 0 λ 1 , we have λx 1 + (1 λ ) x 2 C . The geometric interpretation: for any two points of C , the line segment joining them lies entirely in C . Theorem. A subset of R n is convex iff it contains all the convex combinations of its elements. ESI 6492. Global Optimization Convex Sets and Functions – p. 2
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Convex Sets and Functions Y-Ax is X-Axis B A A convex set A and nonconvex set B ESI 6492. Global Optimization Convex Sets and Functions – p. 3
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Convex Sets and Functions Let F be a family of convex sets. Then the intersection intersectiontext C F C is also a convex set. Let C be a convex set in R n and α a real number. Then the set αC = { x : x = αy, y C } is also convex . Let C 1 , C 2 be convex sets in R n . Then the set C 1 + C 2 = { x : x = x 1 + x 2 , x 1 C 1 , x 2 C 2 } is also convex . ESI 6492. Global Optimization Convex Sets and Functions – p. 4
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Convex Sets and Functions A hyperplane H in R n is a set of the form H = { x R n : c T x = b } , where c R n \ { 0 } and b R . Similarly we define the closed halfspaces H + = { x R n : c T x b } , H = { x R n : c T x b } . It is easy to see that H, H + , H are all convex sets. ESI 6492. Global Optimization Convex Sets and Functions – p. 5
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Convex Sets and Functions Separation theorems are fundamental in convexity and optimization. An important example of such a theorem is the following result: Theorem. Let C R n be a nonempty closed convex set, and let y / C. Then there exist c R n \ { 0 } , b R such that c T y < b and c T x b x C. ESI 6492. Global Optimization Convex Sets and Functions – p. 6
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Convex Sets and Functions An immediate consequence of the above separation theorem is that closed convex sets can be described by a (possibly infinite) family of linear inequalities. Corollary. A nonempty closed convex set C in R n is the intersection of all closed halfspaces containing C . ESI 6492. Global Optimization Convex Sets and Functions – p. 7
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Convex Sets and Functions Intuitively, in order to express C as the intersection of halfspaces, we need only hyperplanes which are nearest to C , i.e. which contain a boundary point of C .
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