barrier - EE236A(Fall 2007-08 Lecture 12 The barrier method...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EE236A (Fall 2007-08) Lecture 12 The barrier method • brief history of interior-point methods • Newton’s method for smooth unconstrained minimization • logarithmic barrier function • central points, the central path • the barrier method 12–1 The ellipsoid method • 1972: ellipsoid method for (nonlinear) convex nondifferentiable optimization (Nemirovsky, Yudin, Shor) • 1979: Khachiyan proves that the ellipsoid method applied to LP has polynomial worst-case complexity • much slower in practice than simplex • very different approach from simplex method; extends gracefully to nonlinear convex problems • solved important open theoretical problem (polynomial-time algorithm for LP) The barrier method 12–2 Interior-point methods early methods (1950s–1960s) • methods for solving convex optimization problems via sequence of smooth unconstrained problems • logarithmic barrier method (Frisch), sequential unconstrained minimization (Fiacco & McCormick), affine scaling method (Dikin), method of centers (Huard & Lieu) • no worst-case complexity theory; (often) worked well in practice • fell out of favor in 1970s new methods (1984–) • 1984 Karmarkar: new polynomial-time method for LP (projective algorithm) • later recognized as closely related to earlier interior-point methods • many variations since 1984; widely believed to be faster than simplex for very large problems (over 10,000 variables/constraints) The barrier method 12–3 Gradient and Hessian differentiable function f : R n → R gradient and Hessian (evaluated at x ): ∇ f ( x ) = ∂f ( x ) ∂x 1 ∂f ( x ) ∂x 2 . . . ∂f ( x ) ∂x n , ∇ 2 f ( x ) = ∂ 2 f ( x ) ∂x 2 1 ∂ 2 f ( x ) ∂x 1 ∂x 2 ··· ∂ 2 f ( x ) ∂x 1 ∂x n ∂ 2 f ( x ) ∂x 2 ∂x 1 ∂ 2 f ( x ) ∂x 2 2 ··· ∂ 2 f ( x ) ∂x 2 ∂x n . . . . . . . . . . . . ∂ 2 f ( x ) ∂x n ∂x 1 ∂ 2 f ( x ) ∂x n x 2 ··· ∂ 2 f ( x ) ∂x 2 n 2nd order Taylor series expansion around x : f ( y ) ≃ f ( x ) + ∇ f ( x ) T ( y − x ) + 1 2 ( y − x ) T ∇ 2 f ( x )( y − x ) The barrier method 12–4 Positive semidefinite matrices a quadratic form is a function f : R n → R with f ( x ) = x T Ax = n summationdisplay i,j =1 A ij x i x j may as well assume A = A T since x T Ax = x T (( A + A T ) / 2) x A = A T is positive semidefinite if x T Ax ≥ for all x A = A...
View Full Document

Page1 / 12

barrier - EE236A(Fall 2007-08 Lecture 12 The barrier method...

This preview shows document pages 1 - 4. Sign up to view the full document.

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