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MATHEMATICS 3 Operational Analysis Štefan Berežný Applied informatics Košice - 2010

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

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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.

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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.)

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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.

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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.

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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.

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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.

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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.

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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 *.

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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.

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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).

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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.

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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.

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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 32 040 02 Košice

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