lecture9 - Problems IE417 Nonlinear Programming Lecture 9...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
lehigh-logo IE417: Nonlinear Programming: Lecture 9 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 14th February 2006 Jeff Linderoth IE417:Lecture 9 lehigh-logo Problems Homework notes... For optimizers, there is no such thing as a positive definite matrix that is non-symmetric ! Hessians are always positive semidefinite x T Ax = 1 / 2 x T ( A + A T ) x , and A + A T is symmetric Jeff Linderoth IE417:Lecture 9 lehigh-logo From the Home Office in Macungie, PA Top Ten Algorithms of the Century 1 Monte Carlo Method 2 Simplex Method 3 Conjugate Gradient Method 4 Matrix Decompositions and Factorizations 5 Optimizing FORTRAN Compiler 6 QR Algorithm 7 Quicksort 8 Fast Fourier Transform 9 Integer Relation Detection Algorithm 10 Fast Multipole Algorithm Jeff Linderoth IE417:Lecture 9 lehigh-logo Today: Conjugate Gradient If A ∈ S n × n ++ , then solving Ax = b is equivalent to min x R n φ ( x ) def = 1 / 2 x T Ax - b T x φ ( x ) = r ( x ) = Ax - b Conjugate Gradient algorithm generates conjugate directions Conjugate Directions Non-zero vectors { d 0 , d 1 , . . . d } are conjugate directions with re- spect to a matrix A if d T i Ad j = 0 i, j Note: A = I orthogonal Jeff Linderoth IE417:Lecture 9
Image of page 1

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

View Full Document Right Arrow Icon
lehigh-logo Conjugate Vectors are Linearly Independent Lemma If { d 1 , . . . d n } are conjugate with respect to A R n × n , then { d 1 , . . . d n } are linearly independent. Proof. If α k = 0 k is the only solution to α 1 d 1 + . . . α n d n = 0 , then { d 1 , . . . d n } are linearly independent. For each k it is simple to establish that α k = 0 by applying the conjugacy property to the system d k A ( α 1 d 1 + . . . α n d n ) = 0 Jeff Linderoth IE417:Lecture 9 lehigh-logo Conjugate Directions Method Let { d 0 , d 1 , . . . , d
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern