LPChap2.doc - 11 The understanding of the solution methods requires the proof of a theorem To follow the proof of the theorem we need to go through some

LPChap2.doc - 11 The understanding of the solution methods...

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The understanding of the solution methods requires the proof of a theorem. To follow the proof of the theorem we need to go through some definitions. These are introduced below: Definition 1.1 . A convex combination of n -dimensional points m U U U ,... , 2 1 is a point m m U U U U ... 2 2 1 1 where i are scalars, m i i i 1 . 1 and 0 1 Let us consider points U 1 = (1, 1) and U 2 = (4, 5) in 2-dimensional case. Then the convex combination of 2 1 and U U is the line segment joining these points as shown by the bold portion of the line in the following figure. Y (4.5) (1, 1) X Figure 1.8 Graphical representation of convex combination of 2 points in a plane The equation of the line passing through (1, 1) and (4,5) is 3y = 4x -1. Let us check it for 3 2 , 3 1 2 1 . Then 3 11 , 3 ) 5 , 4 ( 3 2 ) 1 , 1 ( 3 1 2 2 1 1 U U U , which satisfy the equation of the line. Check that it belongs to the line segment between (1,1) and (4,5). Definition1.2 : A subset C of R n is called a convex set if, for all pair of points 2 1 and U U in C, any convex combination 2 1 ) 1 ( U U U is also in C. That is, if the line segment joining any two points of a set wholly lies in the set, then this set is called a convex set. Thus if we consider the set of all points on a triangle including the boundary points, then this set is a convex set, because the line segment joining any two points of the set also lies in the set. But the set of all boundary points of the triangle is not a convex set (Why?). Examples of convex sets are the whole space R n , a circle, a rectangle, a square and a cube. Some convex and non-convex sets are shown in the following figure: A portion of segments of the lines is outside of the set Non-convex Figure1.9 : Examples of convex and non-convex set 11 Conv exx Conv ex N on - co nv ex
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Definition 1.3: A convex set C is called unbounded if for every point U in C, a point 0 T can be found such that the points T U belongs to C for all . 0 Otherwise C is called bounded . Thus C is bounded if there is a point U in C such that, for every , 0 T there exists a for which T U does not belongs to C. In the following figures bounded and unbounded convex sets are shown by the shaded region and arrow head lines respectively. Y Y X X (a) Bounded convex set (b) Unbounded convex set Figure1.10 : Bounded and unbounded convex sets Definition 1.4. A point U in a convex set C is called an extreme point of the convex set C, if it cannot be expressed as a convex combination of any other two distinct points in C. That is, if U does not lie on a line segment joining any other two distinct points of C. Thus the vertices of a triangle are extreme points. The points on the circumference of a circle are extreme points. Definition1.5 : Now let us define the general Linear Programming Problem ( LPP ) for minimization (maximization) problem. The purpose of an LPP is to find out a vector X ) ,... ,..., , ( 2 1 n j x x x x which minimizes (maximizes) the objective function f ( X ) n n j j x c x c x c x c ... ....
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