lec3 - 15.053 z February 13, 2007 The Geometry of Linear...

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1 15.053 February 13, 2007 z The Geometry of Linear Programs the geometry of LPs illustrated
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Quotes of the day You don't understand anything until you learn it more than one way. Marvin Minsky One finds limits by pushing them. Herbert Simon 2
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Goal of this Lecture z Present the Geometry of Linear Programs A key way of looking at LPs Others are algebraic and economic Some basic concepts 2-dimensional (2 variable) linear programs) 3-dimensional (3 variable) linear programs Properties of the set of feasible solutions and of optimal solutions generalizable to all linear programs 3
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A Two Variable Linear Program (a variant of the DTC example) z = 3x + 5y objective 2x + 3y 10 x + 2y 6 x + y 5 x 4 y 3 x, y 0 (1) (2) (3) (4) (5) (6) We could have used the original variable names of K and S, but it is simpler to use x and y since we usually think of the two axes as the x and y axis. 4
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Finding an optimal solution z Introduce yourself to your partner z Try to find an optimal solution to the linear program, without looking ahead. Finding an optimal solution to a 2-variable LP can be challenging until you have seen the theory. Finding an optimal solution to an LP with more than a few variables and constraints is very hard to do by hand (or at least prone to errors) and we typically use a computer. 5
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6 Some Basic Concepts 1 2 3 4 5 6 1 2 3 4 5 A point is represented as a pair (x, y). For example, (2, 3). x y Sometimes, we will call (x, y) a vector . In that case, it is often represented with a line segment directed from the origin. We go through this review pretty quickly
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7 Lines 1 2 3 4 5 6 1 2 3 4 5 x y Every pair of (distinct) points determines a unique line. p 1 = (1, 5) p 2 = (4, 2) L: x + y = 6. p 1 Alternative representation of the line: (1- λ )p 1 + λ p 2 for - ∞≤λ≤∞ . p 2 L = (1, 5) + λ (3, -3) for - . The alternative representation is really important, as we shall see on the next few slides.
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8 Rays 1 2 3 4 5 6 1 2 3 4 5 x y Every pair of (distinct) points determines a unique ray beginning at the first point. p 1 = (1, 5) p 2 = (4, 2) p 1 Ray: (1- λ )p 1 + λ p 2 for 0 ≤λ≤∞ . = (1, 5) + λ (3, -3) for 0 . p 2
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9 Line segments 1 2 3 4 5 6 1 2 3 4 5 x y Every pair of points determines a unique line segment. p 1 = (1, 5) p 2 = (4, 2) p 1 Segment: (1- λ )p 1 + λ p 2 for 0 ≤λ≤ 1. p 2 = (1, 5) + λ (3, -3) for 0 1. We keep seeing (1- λαμβδα )p 1 + lambda p 2 as the formula. But the representation of the line segment is the most useful for our purposes.
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10 Inequalities 1 2 3 4 5 6 1 2 3 4 5 x y An inequality with two variables determines a unique half-plane x+ 2y 6 A half plane contains the line as well as all points on one side of the line.
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Graphing the Feasible Region We will construct and shade the feasible region one or two constraints at a time. 11 1 2 3 4 5 6 1 2 3 4 5 x y
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12 1 2 3 4 5 6 1 2 3 4 5 Graph the Constraints : 2x+ 3y 10 (1) x 0 , y 0. (6) x y 2x + 3y = 10 OK. This is really three constraints despite what was said on the last slide.
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13 1 2 3 4 5 6 1 2 3 4 5 Add the Constraint : x + 2y 6 (2) x y x + 2y = 6
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Add the Constraint : x + y 5 y 5 y = 5 4 3 2 1 x +
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This note was uploaded on 01/17/2012 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.

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lec3 - 15.053 z February 13, 2007 The Geometry of Linear...

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