Lecture 18 - Linear Programs in Standard Form

Lecture 18 - Linear Programs in Standard Form - Linear...

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Lecture 18 Linear Programs in Standard Form UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Monday, February 23rd, 2009 Linear programs in standard form minimize x R n c T x subject to Ax = b | {z } equality constraints , x 0 | {z } simple bounds The matrix A is m × n with shape A = Often, n ± m . We will show that the two n × n systems A T k λ k = c and A k p k = e s are equivalent to two systems of order m . UCSD Center for Computational Mathematics Slide 2/36, Monday, February 23rd, 2009 Every linear program can be written in standard form. For example, suppose the constraints are in the format Ax b . there are no simple bounds (i.e., the x j are “free variables”) There are two steps involved in reformulating the constraints in standard form. 1 First we convert all the free variables into bounded variables 2 Then we convert the general inequalities into equalities UCSD Center for Computational Mathematics Slide 3/36, Monday, February 23rd, 2009 Define new nonnegative variables u i and v i such that x i = u i - v i , u i 0 , v i 0 Then Ax = A ( u - v ) = Au - Av = ( A - A ) ± u v ² b This gives the linear program with n 0 = 2 n variables: minimize x 0 R n 0 c 0 T x 0 subject to A 0 x 0 b 0 , x 0 0 with A 0 = ( A - A ) , x 0 = ± u v ² , c 0 = ± c - c ² and b 0 = b UCSD Center for Computational Mathematics Slide 4/36, Monday, February 23rd, 2009
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Now we assume that every variable is simply bounded, i.e., x i 0. minimize x 1 , x 2 , x 3 - 6 x 1 - 9 x 2 - 5 x 3 subject to 2 x 1 + 3 x 2 + x 3 5 x 1 + 2 x 2 + x 3 3 x 1 , x 2 , x 3 0 Consider a new variable x 4 and the constraint 2 x 1 + 3 x 2 + x 3 + x 4 = 5 For feasible x 1 , x 2 , x 3 and for x 4 0 , then 2 x 1 + 3 x 2 + x 3 5 x 4 is called a slack variable . UCSD Center for Computational Mathematics Slide 5/36, Monday, February 23rd, 2009 Consider a new variable x 5 and the constraint x 1 + 2 x 2 + x 3 - x 5 = 3 For feasible x 1 , x 2 , x 3 and x 5 0 , then x 1 + 2 x 2 + x 3 3 x 5 is called a surplus variable . UCSD Center for Computational Mathematics Slide 6/36, Monday, February 23rd, 2009 minimize x 1 , x 2 , x 3 , x 4 , x 5 - 6 x 1 - 9 x 2 - 5 x 3 subject to 2 x 1 + 3 x 2 + x 3 + x 4 = 5 x 1 + 2 x 2 + x 3 - x 5 = 3 x 1 , x 2 , x 3 , x 4 , x 5 0 “Slack variable” is a generic name for both slack and surplus variables An equality constraint a T x = b needs neither a slack nor surplus variable. UCSD Center for Computational Mathematics Slide 7/36, Monday, February 23rd, 2009 Optimality conditions for an LP in standard form minimize x R n c T x subject to Ax = b x 0 minimize x R n c T x subject to Ax = b Dx f with ± D = I f = 0 The “full” vector of residuals at a feasible point is: A D ! x
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Lecture 18 - Linear Programs in Standard Form - Linear...

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