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ee227a_review - UC Berkeley Department of Electrical...

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UC Berkeley Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Review 1 Fall 2009 Problem 1.1 Consider the function f : R 2 R given by: f ( x, y ) = ( x 2 y ) 2 10 x 2 . (a) Is f convex or not? (b) Show that f has exactly one stationary point over R 2 , and characterize whether it is a local minimum, a local maximum, or neither. (c) Now consider the constrained optimization of f over the set X = { ( x, y ) R 2 | 0 y 1 } . Show that a global minimum exists, and find all points at which it is attained. Problem 1.2 Consider the inequality-constrained problem min x R n f ( x ) such that x C C = { x R n | g j ( x ) 0 , j = 1 , . . . , m } , where f and g j , j = 1 , . . . m are convex and differentiable functions on R n . Suppose that x * is feasible, and the pair ( x * , μ * ) satisfy the Karush-Kuhn-Tucker necessary conditions, including the complementary slackness condition. Using the convexity and KKT conditions, show that [ f ( x * )] T [ x x * ] 0 for all x C .
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