The Steepest Descent Algorithm for Unconstrained Optimization and ...

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6.3 Modification of the Bisection Algorithm when the Domain of f (x) is Restricted The discussion and analysis of the bisection algorithm has presumed that our optimization problem is P : minimize f (x) n s.t. x ∈ R . Given a point ¯x and a direction d¯, the line-search problem then is LS : minimize h(α) := f (¯x + αd¯) s.t. α ∈ R. n Suppose instead that the domain of definition of f (x) isanopenset X ⊂ R . Then our optimization problem is: P : minimize f (x) s.t. x ∈ X, and the line-search problem then is LS : minimize h(α) := f (¯x + αd¯) s.t. x ¯ + αd ¯∈ X. In this case, we must ensure that all iterate values of α in the bisection alx ¯ + αd ¯ ∈ X. As an example, consider the following gorithm satisfy problem: ∑ m P : minimize f (x) := − ln(b i − A i x) i=1 s.t. b − Ax > 0. 22
Here the domain of f(x) is X = {x ∈R | b − Ax > 0}. Given ¯x ∈ X and a direction d¯, the line-search problem is: n a point ∑ LS : minimize h(α) := f(¯ x + αd¯) = − m ln(b i − A i (¯ x + αd¯)) i=1 s.t. b − A(¯x + αd¯) > 0. Standard arithmetic manipulation can be used to establish that b − A(¯ x + αd¯) > 0 if and only if α
Page 1 and 2: The Steepest Descent Algorithm for
Page 3 and 4: A natural consequence of this is th
Page 5 and 6: ∗ the optimal solution x = x . In
Page 7 and 8: In order for the convergence consta
Page 9 and 10: k x k 1 x k 2 k d 1 k d 2 ||d k ||
Page 11 and 12: 10 8 6 4 2 0 −2 −5 0 5 Figure 2
Page 13 and 14: Figure 3: Hem-stitching in the exam
Page 15 and 16: k x k 1 x k 2 x k 3 x k 4 ||d k ||
Page 17 and 18: • Kantorovich won the Nobel Prize
Page 19 and 20: 20 10 0 −10 −20 f(x ,x−30) 1
Page 21: • If h ′ (˜ α) > 0, re-set α
Page 25 and 26: 1 1 λ 1 a + λ n A = (λ 1a + λ n
Page 27 and 28: • x 0 =(0, 0) T . • x 0 =(−0.

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