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

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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
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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
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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 0 for all x A = A T is positive definite if x T Ax > 0 for all x negationslash = 0 The barrier method 12–5 Convex differentiable functions f : R n R is convex if for all x and y 0 λ
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