L14_uncon - Lecture 14: Algorithms Unconstrained...

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Unformatted text preview: Lecture 14: Algorithms Unconstrained Optimization March 7, 2007 Lecture 14 Outline Terminology and Assumptions Gradient Descent Method Newtons Method Convex Optimization 1 Lecture 14 Unconstrained Minimization minimize f ( x ) Assumptions: The function f is convex and continuously differentiable over dom f The optimal value f * = inf x f ( x ) is finite and attained [ x * exists ] Minimization Methods Produce a sequence of points x k dom f , k = 0 , 1 , . . . , such that f ( x k ) f * Can be interpreted as iterative methods solving optimality condition f ( x * ) = 0 Convex Optimization 2 Lecture 14 Initial Point and Level Set Assumptions Algorithms that we will consider require a starting point x such that It is feasible: x dom f The level set L = { x | f ( x ) f ( x ) } is closed The closedness condition: Not always easy to verify Satisfied when all level sets are closed ; this is guaranteed when: The epigraph epi f of f is closed [ f closed; f lsc] The domain of f is the entire space: dom f = R n f increases to + as the boundary of the domain is approached: f ( x ) as x bd ( dom f ) An example of a differentiable convex function with closed level sets: f ( x ) =- m X i =1 ln( b i- a T i x ) dom f = { x | Ax b } Convex Optimization 3 Lecture 14 Strong Convexity Assumption and Implications In convergence analysis : strong convexity of f is often used. Strong convexity assumption : f is twice continuously differentiable and there exists an m > such that 2 f ( x ) mI for all x L Implications : Lower Bound on f over L : f ( y ) f ( x ) + f ( x ) T ( y- x ) + m 2 k x- y k 2 2 for all x, y L (1) minimize w/r to y in the right-hand side: f ( y ) f ( x )- 1 2 m k f ( x ) k 2 minimum over y L : f ( x )- f * 1 2 m k f ( x ) k 2 Useful as stopping criterion (if you know m ) Relation (1) with x = x and f ( y ) f ( x ) imply that L is bounded Convex Optimization 4 Lecture 14 Upper Bound on Hessian and f over the Level Set For a strongly convex f : The level set L = { x | f ( x ) f ( x ) } is bounded (just shown) The maximum eigenvalue of the Hessian 2 f ( x ) is a continuous function of x over L Hence, the maximum eigenvalue of the Hessian is bounded over L : there is a constant M such that 2 f ( x ) MI for all x L Upper Bound on f over L : f ( y ) f ( x ) + f ( x ) T ( y- x ) + M 2 k y- x k 2 for all x, y L minimize over y L in both sides: f * f ( x )- 1 2 M k f ( x ) k 2 for all x L Convex Optimization 5 Lecture 14...
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L14_uncon - Lecture 14: Algorithms Unconstrained...

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