HW3_sols_2018 - Nonlinear Optimization MATH 408 Winter 2018 Homework 3(Solutions Exercises 2.2 2.4 2.7 2.13 2.15 2.17 in Introduction to Nonlinear

HW3_sols_2018 - Nonlinear Optimization MATH 408 Winter...

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Nonlinear Optimization Homework 3 (Solutions) MATH 408 Winter 2018 Exercises: 2.2, 2.4 - 2.7, 2.13, 2.15, 2.17 in “Introduction to Nonlinear Optimization: Theory, Algorithms, and Applications with MATLAB” Problem 2.2(3 pts) LetaRnbe a nonzero vectpr.Show that the maximum off(x) =aTxoverB={xRn:kxk ≤1}is attained atx*=akakand that the maximalvalue iskak. Problem 2.4(2 pts) Show that ifA, Baren×npositive semidefinite (psd) matrices,thenA+Bis also psd.SupposeA, B0. For all Problem 2.5(3pts) LetARn×nandBRm×mbe two symmetric matrices. Prove thefollowing are equivalent.(1)AandBare psd(2) 1

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