{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture 22 - Taking the Dual

Lecture 22 - Taking the Dual - Recap duality Lecture 22...

This preview shows pages 1–4. Sign up to view the full content.

Lecture 22 Taking the Dual UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Friday, March 6th, 2009 Recap: duality Primal in all-inequality form : (P): minimize x R n c T x subject to Ax b (D): maximize y R m b T y subject to A T y = c , y 0 Primal in standard form : (P): minimize x R n c T x subject to Ax = b , x 0 (D): maximize y R m b T y subject to A T y c UCSD Center for Computational Mathematics Slide 2/33, Friday, March 6th, 2009 Rules for primal-dual conversion Taking the dual ” involves three steps: 1 Convert the given LP to a generic all-inequality or standard form for the primal 2 “Take the dual”. Define the dual problem in terms of the generic quantities 3 Simplify UCSD Center for Computational Mathematics Slide 3/33, Friday, March 6th, 2009 Consider the primal in all-inequality form: minimize x R n c T x subject to Ax b A schematic representation of the problem data is " c T A b # The schematic form of the dual, in standard form , is then " - b T - A T - c # To see this we write down the dual: minimize y R m ( - b ) T y subject to ( - A ) T y = ( - c ) , y 0 UCSD Center for Computational Mathematics Slide 4/33, Friday, March 6th, 2009

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
From the previous slide: minimize y R m ( - b ) T y subject to ( - A ) T y = ( - c ) , y 0 Now we simplify, to give minimize y R m - b T y subject to A T y = c , y 0 which is the same as maximize y R m b T y subject to A T y = c , y 0 UCSD Center for Computational Mathematics Slide 5/33, Friday, March 6th, 2009 Now consider the primal in standard form : minimize x R n c T x subject to Ax = b , x 0 A schematic representation of the problem data is " c T A b # The dual, in all-inequality form , has data " - b T - A T - c # which gives the dual problem: minimize y R m ( - b ) T y subject to ( - A ) T y ( - c ) UCSD Center for Computational Mathematics Slide 6/33, Friday, March 6th, 2009 From the previous slide: minimize y R m ( - b ) T y subject to ( - A ) T y ( - c ) Now we simplify, maximize y R m b T y subject to A T y c UCSD Center for Computational Mathematics Slide 7/33, Friday, March 6th, 2009 Example Next we form the dual of a linear program that is not already in one of the two generic primal forms. Consider the problem minimize w R n d T w subject to Gw f , w 0 UCSD Center for Computational Mathematics Slide 8/33, Friday, March 6th, 2009
From the previous slide is: minimize w R n d T w subject to Gw f , w 0 Convert. We choose to write the primal in all-inequality form G I n w f 0 This problem has data: d T G f I n 0 UCSD Center for Computational Mathematics Slide 9/33, Friday, March 6th, 2009 Example Take the dual.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern