ee227a_review - UC Berkeley Department of Electrical...

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Unformatted text preview: 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 | ≤ 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 ) ≤ , 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....
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This note was uploaded on 09/03/2011 for the course EE 227A taught by Professor Staff during the Spring '11 term at University of California, Berkeley.

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ee227a_review - UC Berkeley Department of Electrical...

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