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Unformatted text preview: UC Berkeley Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Problem Set 6 Fall 2009 Issued: Thursday, November 5 Due: Thursday, November 19, 2009 Problem 6.1 B & V, Problem 5.21 Problem 6.2 B & V, Problem 5.31 Problem 6.3 B & V, Problem 5.39 Problem 6.4 B & V, Problem 5.43 Problem 6.5 Consider a problem of the form p * = min x f ( Ax + b ) + 1 2 bardbl x bardbl 2 2 where f : R m R is a convex function (whose epigraph is a closed set), and A R m n is a given matrix. The purpose of this exercise is to show that the optimal value p * is a convex function of kernel matrix K := AA T R m m . (This fact has important computational consequences for classification and regression problems.) (a) Form a dual for the problem. Hint: introduce extra variables and constraints, and use the conjugate of f in your expression of the dual. (b) Show that strong duality holds, using the result of Exercise 5.25 from BV, and prove that the function p * is convex in K ....
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