(15 points) Let x, E Rm be the j-th column of X E Rmxn. Let y...

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(15 points) Let x, E Rm be the j-th column of X E Rmxn. Let y e Rm and A > 0 be given. Given a vector w E R", define the following function J(w) = ||Xw - y|} + \||w||1. Letting the i-th component w; of w vary and the other components of w be fixed, consider the following one-variable minimization problem reduced from J(w): min f(w;) = min || w;x; – y| A|w;| + 1 lw;| Wi Wi j=1 j#i min ||w;x; +r|| + \w;|+C т min (w,x ji + r;)² + A/w;| + C, j=1 where r = Ejti w,x; – y is in Rm with r = (ra)mx1, and C = XEiti w;l. Show that the optimal solution w for the above minimization problem is given by if lal < 사 if =ta > 0, ta < 0, -A+a Ata if - E 20 jirj and b= E 20;. where a = i=D1 (15 points) Let x, E Rm be the j-th column of X E Rmxn. Let y e Rm and A > 0 be given. Given a vector w E R", define the following function J(w) = ||Xw - y|} + \||w||1. Letting the i-th component w; of w vary and the other components of w be fixed, consider the following one-variable minimization problem reduced from J(w): min f(w;) = min || w;x; – y| A|w;| + 1 lw;| Wi Wi j=1 j#i min ||w;x; +r|| + \w;|+C т min (w,x ji + r;)² + A/w;| + C, j=1 where r = Ejti w,x; – y is in Rm with r = (ra)mx1, and C = XEiti w;l. Show that the optimal solution w for the above minimization problem is given by if lal < 사 if =ta > 0, ta < 0, -A+a Ata if - E 20 jirj and b= E 20;. where a = i=D1

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The minimization problem is given as min fw min Ewx y3 w 1 lw 1 Wi Wi j1 jti min wx r Aw C 2 m m... View the full answer