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secn2 - 2 Linear programs In this section we discuss the...

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2 Linear programs In this section, we discuss the ingredients of linear programs more carefully. As an example for our discussion, here is a simple linear program: (2.1) maximize x 1 + x 2 subject to x 1 2 x 1 + 2 x 2 4 x 1 , x 2 0 . Any linear programming problem involves assigning real-number values to certain variables , which we often denote x 1 , x 2 , . . . , x n . In our example, the integer n is 2. Central to linear programming is the idea of a linear function of these variables, by which we mean a function of the form a 1 x 1 + a 2 x 2 + · · · + a n x n , where the numbers a 1 , a 2 , . . . , a n are given in advance — we call such numbers data . In our example, x 1 +2 x 2 is a linear function. More compactly, we could write our general linear function as n j =1 a j x j , or, using vector notation, as a T x , where T denotes the transpose, a is the column vector [ a 1 , a 2 , . . . , a n ] T and x is the column vector [ x 1 , x 2 , . . . , x n ] T . In a linear program, the values we assign to the variables must satisfy a given list of linear constraints , each of which requires a linear function to be either equal to some given number b (a linear equation ), or be at least b , or at most b ( linear inequalities ). Our example involves four linear inequality constraints: x 1 2, x 1 + 2 x 2 4, x 1 0 and x 2 0. Any assignment of values to the variables that satisfies all the constraints is called a feasible solution . In our simple linear program, x 1 = 1, x 2 = 1 is an example of a feasible solution.
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