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hw4 - 22M:174 Optimization Techniques Homework 4 due Monday...

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22M:174 Optimization Techniques Homework 4 — due Monday Mar 19 1. Suppose that A is a symmetric matrix. This problem is about finding a number τ that guarantees that A + τ I is positive definite. The idea is based on an estimate for eigenvalues called the Gershgorin circle theorem . Show that λ min ( A ) min i a ii - j 6 = i | a i j | ! . [ Hint: Suppose v 6 = 0 and A v = λ v where λ = λ min ( A ) . Then λ v i = n j = 1 a i j v j for all i . Choose i so that | v i | = max j v j . Then use ( λ - a ii ) v i = j 6 = i a i j v j . Take absolute values and use the usual rules for absolute values and v j ≤ | v i | for all j to get the bound on | λ - a ii | .] This bound is often useful for giving a starting point for more refined algo- rithms. 2. Implement the dog-leg trust region method with m k ( p ) = f ( x k )+ p T f ( x k )+ 1 2 p T 2 f ( x k ) p : set p * k to be the minimizer of m k on the line segment joining the Cauchy point p C k to the Newton point p N k = - 2 f ( x k ) - 1 f ( x k ) . Apply this method to the problem of minimizing f ( x , y ) = x 4 - x 2 y + 2 y 2 - 2 y
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