bv_cvxbook_extra_exercises

This bi criterion problem can be formulated as an

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Unformatted text preview: e). Of course, we have chosen an example in R2 so the ellipses can be plotted, but one can detect outliers in R2 simply by inspection. In dimension much higher than 3, however, detecting outliers by plotting will become substantially more difficult, while the same algorithm can be used. Note. In CVX, you should use det_rootn (which is SDP-representable and handled exactly) instead of log_det (which is handled using an inefficient iterative procedure). 73 8 Unconstrained and equality constrained minimization 8.1 Gradient descent and nondifferentiable functions. (a) Let γ > 1. Show that the function f (x1 , x2 ) = x2 + γx2 |x2 | ≤ x1 1 2 x1 + γ | x2 | √ otherwise 1+γ is convex. You can do this, for example, by verifying that f (x1 , x2 ) = sup x1 y1 + √ γx2 y2 2 2 y1 + y2 ≤ 1, y1 ≥ 1/ 1 + γ . Note that f is unbounded below. (Take x2 = 0 and let x1 go to −∞.) (b) Consider the gradient descent algorithm applied to f , with starting point x(0) = (γ, 1) and an exact line search. Show...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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