NLP_Midterm_2003_Sol - 6.252 Spring 2003 Prof D P Bertsekas...

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6.252, Spring 2003, Prof. D. P. Bertsekas Midterm In-Class Exam, Closed-Book, One Sheet of Notes Al- lowed Problem 1: (30 points) (a) Consider the method x k +1 = x k + α k d k for unconstrained minimization of a continuously differentiable function f : n . State which of the following statements are true and which are false. You don’t have to justify your answers: 1. If d k = −∇ f ( x k ) and α k is such that f ( x k +1 ) < f ( x k ) whenever f ( x k ) = 0, every limit point of the generated sequence { x k } is stationary. Solution: False. See Figure 1.2.6 in section 1.2. 2. If d k = −∇ f ( x k ), α k is chosen by the Armijo rule, and the function f has the form f ( x 1 , x 2 ) = ( x 1 ) 2 +( x 2 ) 2 + x 1 the generated sequence { x k } converges to a global minimum of f . Solution: True. We know that every limit point of steepest descent with the Armijo rule converges to a stationary point (Prop. 1.2.1). Since the method is a descent method, all iterates are contained in the level set of the starting point, which is bounded because the cost function is coercive. Hence there is at least one limit point, which must be stationary. Since the cost function is strictly convex, it has unique stationary point which is the global minimum. Therefore, the method converges to this global minimum. (b) Consider the minimization of f ( x ) = x 2 subject to x X where X = { x | x 1 + · · · + x n = 1 } . State which of the following statements are true and which are false. You don’t have to justify your answers: 1. The conditional gradient method with some suitable stepsize rule can be used to obtain a global minimum.
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