a) Write down the augmented Lagrangian function for a given penalty parameter p>0. (5 points) b)...

  

  

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a) Write down the augmented Lagrangian function for a given penalty parameter p>0. (5 points) b) Write down (with derivation) the ADMM update formulas for all variables. (10 points) min B1B2,B3.BERP i=1 s.t. B1 = B, B2 = B, B3 = B where y; E R150 is the i-th batch and X E R150x960 for i = 1,2,3. The algorithm we are going to use is a "block-coordinate" ADMM. Alternating Direction Method of Multipliers (ADMM) Define the augmented Lagrangian function (for a given p) as following: L(x, z, u; p) = f(x) + g(z) + u"(Ax + Bz – c) +||Ax + Bz - Then the ADMM updates are as following: Initiate zo and uo and for k = 1,2, ... iterate using the following: Xk = arg min Lx, Zk-1, Uk-1 ip) Zk = arg min L (Xk, Z, Uk-1 ;p) Uk = Uk-1 + p(Axj + Bzk - c) Note that original augmented Lagrangian combines the updates of x, z: (Xk, Zk) = arg min L(x, z, Uk-1 ;p) x,Z Alternating Direction Method of Multipliers (ADMM) Assume we want to solve the following problem: min f (x) + g(z) subject to Ax + Bz = c X,z Similar to the Augmented Lagrangian method, we augment the objective function as the following: min f (x) + g(z) +5||Ax + Bz – c||, subject to Ax + Bz = c x,z The augmented term does not change the objective, as its value is always zero due to the constraints. Define the augmented Lagrangian function (for a given p) as following: L(x, z, u; p) = f(x) + g(z) + u" (Ax + Bz – c) + |Ax + - ell, Lagrangian Augmented a) Write down the augmented Lagrangian function for a given penalty parameter p>0. (5 points) b) Write down (with derivation) the ADMM update formulas for all variables. (10 points) min B1B2,B3.BERP i=1 s.t. B1 = B, B2 = B, B3 = B where y; E R150 is the i-th batch and X E R150x960 for i = 1,2,3. The algorithm we are going to use is a "block-coordinate" ADMM. Alternating Direction Method of Multipliers (ADMM) Define the augmented Lagrangian function (for a given p) as following: L(x, z, u; p) = f(x) + g(z) + u"(Ax + Bz – c) +||Ax + Bz - Then the ADMM updates are as following: Initiate zo and uo and for k = 1,2, ... iterate using the following: Xk = arg min Lx, Zk-1, Uk-1 ip) Zk = arg min L (Xk, Z, Uk-1 ;p) Uk = Uk-1 + p(Axj + Bzk - c) Note that original augmented Lagrangian combines the updates of x, z: (Xk, Zk) = arg min L(x, z, Uk-1 ;p) x,Z Alternating Direction Method of Multipliers (ADMM) Assume we want to solve the following problem: min f (x) + g(z) subject to Ax + Bz = c X,z Similar to the Augmented Lagrangian method, we augment the objective function as the following: min f (x) + g(z) +5||Ax + Bz – c||, subject to Ax + Bz = c x,z The augmented term does not change the objective, as its value is always zero due to the constraints. Define the augmented Lagrangian function (for a given p) as following: L(x, z, u; p) = f(x) + g(z) + u" (Ax + Bz – c) + |Ax + - ell, Lagrangian Augmented

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We given are min 2 B 2 2 3D such that 150 and is ... View the full answer