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The Chinese University of Hong Kong, ShenzheniDDA·Institute for Data and Decision AnalyticsAndre Milzarek·Fall Semester 2019MDS6106 – Introduction to OptimizationSolutions 2Exercise E2.1 (Definiteness of Matrices):Classify the following matrices and verify whether they are positive definite, positive semidefiniteor indefinite.A1=100040002,A2=112112224,A3=001-120-102,A4=-1010-10-10-1.What are the eigenvalues of the matricesA3andA4?
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Finally, let us calculate the eigenvalues ofA4:det(A4-λI) =-(1 +λ) det-1-λ00-1-λ-0 + det0-1-λ-10=-(1 +λ)3-(1 +λ) =-(1 +λ)(2 + 2λ+λ2).Hence, the eigenvalues ofA4areλ1=-1,λ2=-1 +i, andλ3=-1-i, wherei=-1is theimaginary number. In this case,A4does not have real eigenvalues! Let us now check definitenessofA4:xA4x=x1x2x3x3-x1-x2-x1-x3=-(x21+x22+x23).This implies thatA4is negative definite.Exercise E2.2 (Visualization: Sets, Problems, and Contours):In this exercise, we visualize different feasible sets and solve optimization problems via graphicalconsiderations.a) Sketch the following sets inR2:X1:={xR2:x10,|x2| ≤x31},X2:={xR2:x1[0,2], x2[0,2], x1x2(x1+x2-2) = 0},X3:={xR2:x1, x2[-1,1],(x1-1)2+x221, x1≥ -1-x22}.Analyze which of the setsX1,X2, andX3is bounded and closed and explain your answer.b) Consider the nonlinear programminf(x)s.t.xX3(1)and determine (graphically) all local and global minimizer of problem(1)for the two choicesf(x) :=max{x1,0}+x22andf(x) := (12-x1)max{x1,0} -x22. Are the minimizer isolated?c) Write aMATLABorPythoncode to visualize and createa 3D-plot (surface plot) of the functionsf1:R2R, f1(x) = (x21-x2)2+ (x1-x22)2-(x1-1)3+ (x2-1)3,f2:R2R, f2(x) = cos(x1) sin(x2)-x11 +x22.a (two-dimensional) contour plot of the functionsf1andf2.

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Term
Winter
Professor
Prof.Chau
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