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MATHEMATICS 3 Operational Analysis Štefan Berežný Applied informatics Košice - 2010.

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Presentation on theme: "MATHEMATICS 3 Operational Analysis Štefan Berežný Applied informatics Košice - 2010."— Presentation transcript:

1 MATHEMATICS 3 Operational Analysis Štefan Berežný Applied informatics Košice

2 Štefan Berežný Lecture For Applied Informatics 2 Table Of Contents Convex Combination Convex Set Extreme points Corners Basic Feasible Solution Optimal solution

3 Štefan Berežný Lecture For Applied Informatics 3 Convex Analysis Consider any two points A nad B. Then the point C =.A + (1 – ).B, for  0, 1  lies on the line segment joining points A and B. Definition: Given n vectors v 1,..., v n. Vector v is called a convex combination of vectors v 1,..., v n.

4 Štefan Berežný Lecture For Applied Informatics 4 Convex Analysis Definition: A set of points S is called convex if for any subset X of S and for any point P wich we get by convex combination of points in X: P  S. (A set of points S is called convex if for any two points A and B in S the segment joining the points A and B is in X, where X  S and card(X) > 1.)

5 Štefan Berežný Lecture For Applied Informatics 5 Convex Analysis It is easy to see, that the set {x  R n : Ax  b} is convex. This is because for any x and y satysfying: Ax  b and Ay  b, A( x + (1 – )y) = = Ax + (1 – )Ay  b + (1 – )b = b. Theorem: If S and T are two convex sets, then S  T is a convex set.

6 Štefan Berežný Lecture For Applied Informatics 6 Convex Analysis Definition: A point x 0 in a set S is said to be a local maxima for a function f if there exists a small neighbourhood N of x 0 where f(x 0 )  f(x);  x  N. For us, N is a ball of a small but non- zero radius around x 0. Theorem: Let f be a linear function over a convex set S. Then a local maximum is a global maximum.

7 Štefan Berežný Lecture For Applied Informatics 7 Convex Analysis Definition: Given points P 1, P 2, P 3,..., P n, the convex hull (convex envelope, convex closure) is the smallest convex set containing these points. Convex hull of points P 1, P 2, P 3,..., P n is the set of all points which can be written as convex combination of points P 1, P 2, P 3,..., P n.

8 Štefan Berežný Lecture For Applied Informatics 8 Convex Analysis Definition: An extreme point is a point in a convex set that cannot be represented as a convex combination of any two distinct points in the convex set. Theorem: Let P 1, P 2, P 3,..., P n be extreme points of {x  R n : Ax  b}. Then every point in x: Ax  b can be expressed as a convex combination of the points P 1, P 2, P 3,..., P n.

9 Štefan Berežný Lecture For Applied Informatics 9 Convex Analysis Theorem: Every bounded, closed, non-empty and convex set contains at least one extreme point. Theorem: Let set M is convex, closed, bounded and non-empty, then every point of set M can be expressed as convex combination of extreme points of the set M. Theorem: A set of feasible solution to LP problem is convex. Theorem: A set of optimal solutions to LP problem is convex.

10 Štefan Berežný Lecture For Applied Informatics 10 Convex Analysis Theorem: A linear function on convex set S = {x  R n : Ax  b} is minimized at an extreme point. Theorem: Let set M =  x  R n ; A.x = b  x  0  is bounded, non-empty and let c T.x function is linear and defined on the set M. Then: 1) there is a min x  M c T.x = f* 2) there is a extreme point x 0 in set M such that c T.x 0 = f *.

11 Štefan Berežný Lecture For Applied Informatics 11 Convex Analysis Theorem: (Main theorem of LP) Let set M =  x  R n ; A.x = b  x  0  is that c T.x objective function is bounded from below. Then the LP problem c T.x  min over x  M has an optimal solution in some extreme point from set M.

12 Štefan Berežný Lecture For Applied Informatics 12 Convex Analysis Basic Solution Vector x of (Ax = b) is a basic solution if the n components of x can be partitioned into m "basic" and n – m "non-basic" variables in such a way that: - the m columns of A corresponding to the basic variables form a nonsingular basis and - the value of each "non-basic" variable is 0. The constraint matrix A has m rows (constraints) and n columns (variables).

13 Štefan Berežný Lecture For Applied Informatics 13 Convex Analysis Basis The set of basic variables. Basic Variables A variable in the basic solution (value is not 0). Nonbasic Variables A variable not in the basic solution (value = 0). Slack Variable A variable added to the problem to eliminate less-than constraints.

14 Štefan Berežný Lecture For Applied Informatics 14 Convex Analysis Surplus Variable A variable added to the problem to eliminate greater-than constraints. Artificial Variable A variable added to a linear program in phase 1 to aid finding a feasible solution. Unbounded Solution For some linear programs it is possible to make the objective arbitrarily small (without bound). Such an LP is said to have an unbounded solution.

15 Štefan Berežný Lecture For Applied Informatics 15 Thank you for your attention Štefan Berežný Department Of Mathematics and Theoretical Informatics FEI TU Košice B. Němcovej Košice


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