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Unformatted text preview: Lecture 19: Convex NonSmooth Optimization April 2, 2007 Lecture 19 Outline • Convex nonsmooth problems • Examples • Subgradients and subdifferentials • Subgradient properties • Operations with subgradients and subdifferentials Convex Optimization 1 Lecture 19 ConvexConstrained Nonsmooth Minimization minimize f ( x ) subject to x ∈ C • Characteristics : • The function f : R n 7→ R is convex and possibly nondifferentiable • The set C ⊆ R n is nonempty and convex • The optimal value f * is finite • Our focus here is nondifferentiability Renewed interest comes from largescale problems and the need for dis tributed computations. Main questions: • Where do such problems arise? • How do we deal with nondifferentiability? How can we solve them? Convex Optimization 2 Lecture 19 Where they arise • Naturally in some applications (comm. nets, data fitting, neuralnets): • Leastsquares problems minimize ∑ m j =1 k h ( w,x j ) y j k 2 subject to w here ( x j ,y j ) , j = 1 ,...,m are the inputoutput pairs, w are weights (decision variables) to be optimized, h is convex possibly nonsmooth • In Lagrangian duality minimize q ( μ,λ ) subject to μ • A systematic approach for generating primal optimal bounds • A part of some primaldual scheme • In (sharp) penalty approaches min x ∈ C { f ( x ) + tP ( g ( x )) } where t > is a penalty parameter and the penalty function is P ( u ) = ∑ m j =1 max { u j , } or P ( u ) = max { u 1 ,...,u m , } Convex Optimization 3 Lecture 19 Example: Optimization in Network Coding Linear Cost Model minimize ∑ ( i,j ) ∈ L a ij max s ∈ S x s ij subject to ≤ max s ∈ S x s ij ≤ c ij for all ( i,j ) ∈ L ∑ { j  ( i,j ) ∈ L } x s ij ∑ { j  ( j,i ) } x s ji = b s i for all i ∈ N , s ∈ S • N is the set of nodes in the communication network • S is the set of sessions [a session is a pair of nodes that communicate] • L is the set of directed links • ( i,j ) denotes a link originating at node i and ending at node j • c ij is the communication rate capacity of the link ( i,j ) • a ij is the cost for the link ( i,j ) • x s ij...
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 Spring '07
 AngeliaNedich
 Derivative, Convex function, Convex Optimization, Convex analysis

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