# lecture 3 - Math 482 (Lecture 3): Solving 2 variable LP's;...

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Math 482 (Lecture 3): Solving 2 variable LP's; Convexity Today: We are going to discuss a baby version of Section 2.3, and also discuss the notion of convexity (Section 1.5). In particular, we will solve two variable LP's graphically. * Pretty soon, we're going to deal with high multidimensionality, which we navigate through the simplex method. However, in order to better appreciate this method, I believe it is helpful to look at the simplest cases (at most 2 dimensions) geometrically. Since many of the ideas extend. In class exercise: Solve the following two variable problem: Maximize f(x,y)=3x+2y Subject to: 2x+y<= 18 2x+3y<= 42 3x+y<= 24 x,y>=0 Hint/idea: (1) Draw the feasible region on the x-y plane. (2) Determine the "corners" of the feasible region. (3) One of the corners is the maximum. (Why?) -- This is called the corner principle . Solution: See here for (1) and (2). The why of (3) is the main point: consider the "level set" M=3x+2y for a fixed M. As M is made larger, you get a whole line's worth of feasible solutions with objective value M. You can keep on increasing M little by little until you either reach a corner, or a facet

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## lecture 3 - Math 482 (Lecture 3): Solving 2 variable LP's;...

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