# 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
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
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
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
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.
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.
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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