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Unformatted text preview: UC Berkeley Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Solutions 6 Fall 2009 Solution 6.1 (a) p = 1 (b) The Lagrangian is L ( x,y, ) = e x + x 2 /y . The dual function is g ( ) = inf x,y> ( e x + x 2 /y ) = braceleftbigg if otherwise so we can write the dual problem as maximize subject to with optimal value d = 0. The optimal duality gap is p d = 1 (c) Slaters condition is not satisfied. (d) p ( u ) = 1 if u = 0 ,p ( u ) = 0 if u > 0 and p ( u ) = if u < Solution 6.2 Suppose x is feasible. Since f i are convex and f i ( x ) 0, we have f i ( x ) f i ( x ) + f i ( x ) T ( x x ) ,i = 1 ,...,m Using i 0, we conclude that m summationdisplay i =1 i ( f i ( x ) + f i ( x ) T ( x x ) = m summationdisplay i =1 i ( f i ( x ) + m summationdisplay i =1 f i ( x ) T ( x x ) = f ( x ) T ( x x ) In the last line, we use the complementary slackness condition i f i ( x ) = 0, and the last KKT condition. This show that f ( x ) T ( x x ) 0, i.e. f ( x ) defines a supporting hyperplane to feasible set at x Solution 6.3 (a) Follows from tr( Wxx T ) = x T Wx and ( xx T ) ii = x 2 i (b) It gives a lower bound because we minimize the same objective function over a larger set. If X is rank one, it is optimal. 1 (c) We write the problem as a minimization problem minimize 1 T subject to W + diag ( ) followsequal Introducing a Lagrange multiplier X S n for the matrix inequality, we obtain the Lagrangian L ( ,X ) = 1 T tr( X ( W + diag ( ))) = 1 T tr( XW ) n i =1 i X ii = tr( XW ) + n i =1 i (1 X ii ) This is bounded below as a function of only if X ii = 1 for all i , so we obtain the dual problem maximize tr( WX ) subject to X followsequal X ii = 1 ,i = 1 ,...,n Changing the sign again, and switching from maximization to minimization, yields the problem in part (a) Solution 6.4 (a) We introduce the new variables, and write the problem as minimize c T x subject to bardbl y i bardbl 2 t i ,i = 1 ,...,m y i = A i x + b i ,i = 1 ,...,m t i = c T i x + d i ,i = 1 ,...,m The Lagrangian is L ( x,y,t,,, ) = c T x + m summationdisplay...
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