# qp - Quadratic Programming Thank Dusty Sargent for...

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2 Outline x Linearly constrained minimization b Linear equality constraints b Linear inequality constraints x Quadratic objective function
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

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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
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

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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
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qp - Quadratic Programming Thank Dusty Sargent for...

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