λw+νNote that neitherc= 0 orr= 0 allows a solution.Thusμ=ν= 0 by complementary slackness.Moreover,r=Timpliesc= 0, so it is also impossible.Thusρ= 0 by complementary slackness.However,λ= 1/pc >0, sopc+rw=wTis the only binding constraint.Constraint qualification is clearly satisfied with this one constraint. We solve the first-order equa-tions, obtainingc=2wT3pandr=T3.Finally, we need only look at the determinant of the bordered Hessian because there is one constraintand two unknowns. The bordered Hessian isH=0pwp-c-20w0-2r-2.Its determinant is 2(p/r)2+4(cw)2>0. Sincen= 2, detH(-1)2>0, which implies we have a maximum.5. LetAbe ann×npositive definite matrix. DefineN(x) =√xTAx. IsNa norm? That is, does it obey thethree conditions a norm must obey?Answer:It is a norm.1) It is absolutely homogeneous of degree 1 by construction.2) It is clearlynon-negative, and ifN(x) = 0, the quadratic formxTAx= 0. Since this form is positive definite,x= 0.Finally, the triangle inequality must be shown.
This is the end of the preview.
access the rest of the document.