Transcription of Home Assignment 1
1 Home Assignment 1 ECE 602 Introduction to OptimizationDue: January 28, 2022 Exercise 1(Gradient)Letx RnandA Rm n. Also, letf:Rn Rbe defined according tof(x) =m i=1 (Ax)2i+ ,where(Ax)idenotes theith element ofAxand0< 1is a small the gradient off(x)using itsexternal definition. Exercise 2(Convexity)Explain which of the following sets are convex. Show your )The sublevel set of a convex functionf, ,C ={x Rn|f(x) }.b)The set of positive semidefinite matricesSn+.Explain which of the following functions are convex. Show your )f(x) =12xTQx+cTx, whereQ Sn+andc )f(x) =g(h(x))whereh:Rn Ris convex, whileg:R Ris convex andmonotonically increasing1 Exercise 3(Global minimum of convex functions)Letf(x)be a convex function, and letUbe a set of itsglobal minimizers, ,U={x|f(x) f(y), y domf}.
2 Prove thatUis a convex set. [Hint:Use the result in Exercice ( ).] Exercise 4(Dual norms)Prove the following statements:a)The dual norm of||x||1is||x|| .b)The dual norm of||x||2is||x||
