qp - Quadratic Programming Thank Dusty Sargent for...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Quadratic Programming Quadratic Programming Thank Dusty Sargent for providing these slides
Background image of page 1

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

View Full DocumentRight Arrow Icon
2 Outline x Linearly constrained minimization b Linear equality constraints b Linear inequality constraints x Quadratic objective function
Background image of page 2
3 SideBar: Matrix Spaces x Four fundamental subspaces of a matrix b Column space, col(A) b Row space, row(A) b Null space Ax=0, null(A) b Left Null space x T A=0, lnull(A) b Rank =dim(col(A))=dim(row(A)) b Dim(col(A))+Dim(lnull(A)) = # column b col(A) ad lnull(A) are orthogonal b Dim(row(A))+Dim(null(A)) =# row b row(A) and null(A) are orthogonal
Background image of page 3

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

View Full DocumentRight Arrow Icon
4 Linear Equality Constraints x min x F(x) b s.t. Ax = b x Assume constraints are consistent and linearly independent x t contraints remove t degrees of freedom from solution x x x = A T x a + Zx z Row space Null space
Background image of page 4
5 Feasible Search Directions x Feasible points x 1 , x 2 have Ax 1 = Ax 2 = b x Feasible step p satisfies Ap = A(x 1 – x 2 ) = 0 x If Z is a basis for null(A), feasible directions p are such that p = Zp z x I.e., direction of change (p) should be in the null space of A b Ap=0 b Ax 2 =A(x 1 +p) = Ax 1 =b
Background image of page 5

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

View Full DocumentRight Arrow Icon
6 Optimality Conditions x Taylor series expansion along feasible direction b F(x + є Zp z ) = F(x) + є p z T Z T g(x) + ½ є 2 p z T Z T G(x + є u Zp z )Zp z x g is the gradient [f 1 ,f 2 ,…, f n ] T x є p z T Z T g(x) = feasible direction * gradient = change x Projected gradient p
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 19

qp - Quadratic Programming Thank Dusty Sargent for...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online