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Ch8convexsets

# Ch8convexsets - CHAPTER VIII CONVEX SETS We present some...

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CHAPTER VIII CONVEX SETS We present some geometric aspects of linear programming. Definition. A subset S of n is said to be convex, if given x and y in S, then the line segment [ x,y ] connecting x and y lies in S. The line segment [ x,y ] connecting x and y is the set of all points [ x,y ] = { λ x + (1 - λ ) y | 0 λ 1}. Picture of a Convex Set. Example (1). Let A be an m x n matrix and let b be in n . Then the set S = { x n | A x = b } is a convex set. In fact, we have that A x = b and A y = b implies that A( λ x + (1 - λ ) y ) = λ A x + (1 - λ )A y = t b + (1 - λ ) b = b, for 0 λ 1. Thus, [ x, y ] S. Example (2). Let A be an m x n matrix and let b be in n . Then the set S = { x n | A x = b, x 0 } is a convex set. As in Example (1), we have that A x = b and A y = b implies that A( λ x + (1 - λ ) y ) = λ A x + (1 - λ )A y = t b + (1 - λ ) b = b. In addition we have that λ x + (1 - λ ) y 0, for 0 λ 1. Thus, [ x, y ] S. Example (3). Let A be an m × n matrix and let b be in n . Then the set S = { x n | A x b } is a convex set since A( λ x + (1 - λ ) y ) = λ A x + (1 - λ )A y λ b + (1 - λ ) b = b, for 0 λ 1. Definition. A real-valued function f on a convex set S is said to be convex if f( λ x + (1 - λ ) y ) λ f( x ) + (1 - λ )f( y ), for all x and y in S. Graph of a Real Valued Convex Function. Chapter VIII Convex Sets page 1

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Example of a Convex Function. Let c n and let f( x ) = c x (dot product). Then f is a convex function. In fact, the function f is linear and every linear function is convex. Proposition. Let f be a convex function on a convex set S; then every local minimum for f is a global minimum. Proof.Let f( x o ) be a local minimum for f. By the definition of local minimum there is a d > 0 such that f( x o ) f( x ) for every x in S with || x -x o || < δ . Now, for arbitrary y in S, we need to show that f( x o ) f( y ) to show that f( x o ) is a global minimum. We use the convexity of S for this. In fact, we have that || x o - ( λ y + (1 - λ ) ) x o )|| | λ | || x o - y || and thus that || x o - (t y + (1-t) x o )|| < d whenever 0 < λ < δ /(|| x o || + || y || + 1). Thus, we have that f( x o ) f( λ y + (1 - λ ) x o ) λ f( y ) +(1 - λ )f( x o ), or equivalently, tf( x o ) tf( y ) whenever 0 < t < d/(|| x o || + || y || + 1). Thus, f( x o ) f( y ). Q.E.D. The Proposition shows that every local minimum in a Linear Programming Problem is a global minimum. Definition. Let S be a convex set in n ; then a point x in S is said to be an extreme point of S if given y and z in S with x [ y, z ], then x = y = z. Note that x is an extreme point of S if it is not properly inside any line segment in S. Examples of Extreme Points. Chapter VIII Convex Sets page 2
The extreme points of the Triangle The extreme points of the circle are the vertices of the triangle. is the boundary of the circle . Definition. A set of the form H = { x n | c x = a} is said to be a hyperplane in n . Here c is a nonzero vector in n and c x represents the dot product of c and x .

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