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handout20 - Math 171A: Linear Programming Lecture 20 LPs...

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Unformatted text preview: Math 171A: Linear Programming Lecture 20 LPs with Mixed Constraint Types Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Wednesday, February 23rd, 2011 Recap: LP formulations Problems considered so far: ELP minimize x R n c T x subject to Ax = b LP minimize x c T x subject to Ax b Now we consider a mixture of constraint types . UCSD Center for Computational Mathematics Slide 2/32, Wednesday, February 23rd, 2011 Linear programs with mixed constraints minimize x R n c T x subject to Ax = b | {z } equality constraints , Dx f | {z } inequality constraints The dimensions are: A m n b m-vector D m D n f m D-vector UCSD Center for Computational Mathematics Slide 3/32, Wednesday, February 23rd, 2011 Optimality conditions Example: Consider inequalities and just one equality constraint: minimize x R n c T x subject to a T x = b , Dx f Write the equality constraint as two inequalities: a T x b and a T x b , which is equivalent to (- a ) T x - b If x * is an optimal solution, then both a T x b and (- a ) T x - b must be active at x * . UCSD Center for Computational Mathematics Slide 4/32, Wednesday, February 23rd, 2011 Optimality conditions Suppose that x * is a solution of the mixed-constraint problem: minimize x R n c T x subject to a T x b , (- a ) T x - b , Dx f This problem has all inequalities, so we can use existing theory. Let D a denote the matrix of active inequalities at x * . The full active set for the mixed-constraint problem is a T- a T D a x * = b- b f a UCSD Center for Computational Mathematics Slide 5/32, Wednesday, February 23rd, 2011 Optimality conditions From the previous slide: A a = a T- a T D a The optimality conditions A T a a = c , with a 0 are: c = a- a D T a * 1 * 2 z * a , with * 1 , * 2 0 and z * a UCSD Center for Computational Mathematics Slide 6/32, Wednesday, February 23rd, 2011 Optimality conditions From the previous slide: c = a- a D T a * 1 * 2 z * a , with * 1 , * 2 0 and z * a c = a * 1- a * 2 + D T a z * a = a ( * 1- * 2 | {z } positive or negative ) + D T a z * a = a * + D T a z * a , with * = * 1- * 2 UCSD Center for Computational Mathematics Slide 7/32, Wednesday, February 23rd, 2011 Optimality conditions From the previous slide: c = a * + D T a z * a , with z * a * is the Lagrange multiplier for the constraint a T x = b ....
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This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.

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handout20 - Math 171A: Linear Programming Lecture 20 LPs...

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