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# hw2 - EE364b Prof S Boyd EE364b Homework 2 1 Subgradient...

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EE364b Prof. S. Boyd EE364b Homework 2 1. Subgradient optimality conditions for nondifferentiable inequality constrained optimiza- tion. Consider the problem minimize f 0 ( x ) subject to f i ( x ) 0 , i = 1 ,...,m, with variable x R n . We do not assume that f 0 ,...,f m are convex. Suppose that ˜ x and ˜ λ followsequal 0 satisfy primal feasibility, f i x ) 0 , i = 1 ,...,m, dual feasibility, 0 ∂f 0 x ) + m summationdisplay i =1 ˜ λ i ∂f i x ) , and the complementarity condition ˜ λ i f i x ) = 0 , i = 1 ,...,m. Show that ˜ x is optimal, using only a simple argument, and definition of subgradient. Recall that we do not assume the functions f 0 ,...,f m are convex. 2. Optimality conditions and coordinate-wise descent for 1 -regularized minimization. We consider the problem of minimizing φ ( x ) = f ( x ) + λ bardbl x bardbl 1 , where f : R n R is convex and differentiable, and λ 0. The number λ is the regularization parameter, and is used to control the trade-off between small f and small bardbl x bardbl 1 . When 1

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hw2 - EE364b Prof S Boyd EE364b Homework 2 1 Subgradient...

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