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Lect03

# Lect03 - D Bertsekas EE553 LECTURE 3 LECTURE OUTLINE...

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EE553 LECTURE 3 LECTURE OUTLINE Iterative Computational Methods Gradient Methods – Motivation Principal Gradient Methods Gradient Methods - Choices of Direction Gradient Methods - Choice of Stepsize Gradient Methods - Convergence Issues D. Bertsekas Adapted for USC EE553 by M. Safonov © D. Bertsekas, MIT, Cambridge, MA

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EXISTENCE OF OPTIMAL SOLUTIONS Consider min x X f ( x ) Two possibilities: The set ± f ( x ) | x X ² is unbounded below, and there is no optimal solution The set ± f ( x ) | x X ² is bounded below - A global minimum exists if f is continuous and X is compact (Weierstrass theorem) - A global minimum exists if X is closed, and f is coercive, that is, f ( x ) → ∞ when k x k → Adapted for USC EE553 by M. Safonov © D. Bertsekas, MIT, Cambridge, MA
GRADIENT METHODS - MOTIVATION If f ( x ) 6 = 0, there is an interval (0 ) of stepsizes such that f ( x - α f ( x ) ) < f ( x ) for all α (0 ). f(x) = c 1 f(x) = c 2 < c 1 f(x) = c 3 < c 2 x f(x) d x + δ d x α = x + α d If d makes an angle with f ( x ) that is greater than 90 degrees, f ( x ) 0 d < 0 , there is an interval (0 , δ ) of stepsizes such that f ( x + αd ) < f ( x ) for all α (0 ).

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Lect03 - D Bertsekas EE553 LECTURE 3 LECTURE OUTLINE...

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