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# handout20 - Recap LP formulations Math 171A Linear...

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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

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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 λ a = c , with λ a 0 are: c = a - a D T a λ * 1 λ * 2 z * a , with λ * 1 0 , λ * 2 0 and z * a 0 UCSD Center for Computational Mathematics Slide 6/32, Wednesday, February 23rd, 2011 Optimality conditions From the previous slide: c = a - a D T a λ * 1 λ * 2 z * a , with λ * 1 0 , λ * 2 0 and z * a 0 c = a λ * 1 - a λ * 2 + D T a z * a = a ( λ * 1 - λ * 2 | {z } positive or negative ) + D T a z * a = a π * + D T a z * a , with π * = λ * 1 - λ * 2 UCSD Center for Computational Mathematics Slide 7/32, Wednesday, February 23rd, 2011 Optimality conditions From the previous slide: c = a π * + D T a z * a , with z * a 0 π * is the Lagrange multiplier for the constraint a T x = b .
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handout20 - Recap LP formulations Math 171A Linear...

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