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| SDPT3 Version 4.0 | ||
| A MATLAB software for semidefinite-quadratic-linear programming | ||
| Copyright (c) 1997 by | ||
| Kim-Chuan Toh, Michael J. Todd, and Reha H. Tutuncu | ||
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| If you find this software useful for your work, please cite the | ||
| followings: | ||
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| [1] K.C. Toh, M.J. Todd, and R.H. Tutuncu, | ||
| SDPT3 --- a Matlab software package for semidefinite programming, | ||
| Optimization Methods and Software, 11 (1999), pp. 545--581. | ||
| [2] R.H Tutuncu, K.C. Toh, and M.J. Todd, | ||
| Solving semidefinite-quadratic-linear programs using SDPT3, | ||
| Mathematical Programming Ser. B, 95 (2003), pp. 189--217. | ||
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| @@ -1,49 +1,53 @@ | ||
| %%************************************************************************* | ||
| %% ToeplitzApprox: find the nearest symmetric positive definite Toeplitz | ||
| %% matrix to a given symmetric matrix F. | ||
| %% | ||
| %% | ||
| %% max -y(n+1) | ||
| %% s.t. T(y(1:n)) + y(n+1)*B >= 0 | ||
| %% [I 0 ] + sum_{k=1}^n y(k) [0 gam(k)*e_k ] + y(n+1)*B >= 0 | ||
| %% [0 -beta] [gam(k)*e_k' -2q(k) ] | ||
| %% | ||
| %% where B = diag([zeros(n,1); 1]) | ||
| %% q(1) = - Tr(F); q(k+1) = -sum of upper and lower kth diagonals of F | ||
| %% gam(1) = sqrt(n); gam(k) = sqrt(2*(n-k+1)) for k=2:n | ||
| %% beta = norm(F,'fro')^2 | ||
| %%************************************************************************* | ||
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| function [blk,At,C,b] = ToeplitzApprox(F) | ||
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| n = length(F); | ||
| gam = sqrt([n, 2*(n-1:-1:1)]); | ||
| q = zeros(n,1); | ||
| q(1) = -sum(diag(F)); | ||
| for k=1:n-1 | ||
| q(k+1) = -2*sum(diag(F,k)); | ||
| end | ||
| beta = norm(F,'fro')^2; | ||
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| blk{1,1} = 's'; blk{1,2} = n+1; | ||
| blk{2,1} = 's'; blk{2,2} = n+1; | ||
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| b = [zeros(n,1); -1]; | ||
| C{1,1} = sparse(n+1,n+1); | ||
| C{2,1} = spdiags([ones(n,1); -beta],0,n+1,n+1); | ||
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| Acell = cell(1,n+1); | ||
| Acell{1} = -spdiags([ones(n,1); 0],0,n+1,n+1); | ||
| for k = 1:n-1 | ||
| tmp = -spdiags([ones(n,1); 0],k,n+1,n+1); | ||
| Acell{k+1} = tmp + tmp'; | ||
| end | ||
| Acell{n+1} = -spconvert([n+1,n+1,1]); | ||
| At(1,1) = svec(blk(1,:),Acell,1); | ||
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| for k = 1:n | ||
| Acell{k} = -spconvert([k, n+1, gam(k); n+1, k, gam(k); n+1, n+1, -2*q(k)]); | ||
| end | ||
| Acell{n+1} = -spconvert([n+1,n+1,1]); | ||
| At(2,1) = svec(blk(2,:),Acell,1); | ||
| %%*********************************************************************** | ||
| %%************************************************************************* | ||
| %% ToeplitzApprox: find the nearest symmetric positive definite Toeplitz | ||
| %% matrix to a given symmetric matrix F. | ||
| %% | ||
| %% max -y(n+1) | ||
| %% s.t. T(y(1:n)) + y(n+1)*B >= 0 | ||
| %% [I 0 ] + sum_{k=1}^n y(k) [0 gam(k)*e_k ] + y(n+1)*B >= 0 | ||
| %% [0 -beta] [gam(k)*e_k' -2q(k) ] | ||
| %% | ||
| %% where B = diag([zeros(n,1); 1]) | ||
| %% q(1) = - Tr(F); q(k+1) = -sum of upper and lower kth diagonals of F | ||
| %% gam(1) = sqrt(n); gam(k) = sqrt(2*(n-k+1)) for k=2:n | ||
| %% beta = norm(F,'fro')^2 | ||
| %%***************************************************************** | ||
| %% SDPT3: version 4.0 | ||
| %% Copyright (c) 1997 by | ||
| %% Kim-Chuan Toh, Michael J. Todd, Reha H. Tutuncu | ||
| %% Last Modified: 16 Sep 2004 | ||
| %%***************************************************************** | ||
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| function [blk,At,C,b] = ToeplitzApprox(F) | ||
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| n = length(F); | ||
| gam = sqrt([n, 2*(n-1:-1:1)]); | ||
| q = zeros(n,1); | ||
| q(1) = -sum(diag(F)); | ||
| for k=1:n-1 | ||
| q(k+1) = -2*sum(diag(F,k)); | ||
| end | ||
| beta = norm(F,'fro')^2; | ||
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| blk{1,1} = 's'; blk{1,2} = n+1; | ||
| blk{2,1} = 's'; blk{2,2} = n+1; | ||
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| b = [zeros(n,1); -1]; | ||
| C{1,1} = sparse(n+1,n+1); | ||
| C{2,1} = spdiags([ones(n,1); -beta],0,n+1,n+1); | ||
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| Acell = cell(1,n+1); | ||
| Acell{1} = -spdiags([ones(n,1); 0],0,n+1,n+1); | ||
| for k = 1:n-1 | ||
| tmp = -spdiags([ones(n,1); 0],k,n+1,n+1); | ||
| Acell{k+1} = tmp + tmp'; | ||
| end | ||
| Acell{n+1} = -spconvert([n+1,n+1,1]); | ||
| At(1,1) = svec(blk(1,:),Acell,1); | ||
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| for k = 1:n | ||
| Acell{k} = -spconvert([k, n+1, gam(k); n+1, k, gam(k); n+1, n+1, -2*q(k)]); | ||
| end | ||
| Acell{n+1} = -spconvert([n+1,n+1,1]); | ||
| At(2,1) = svec(blk(2,:),Acell,1); | ||
| %%*********************************************************************** |
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| @@ -1,42 +1,47 @@ | ||
| %%************************************************************************* | ||
| %% ToeplitzApproxSQQ: find the nearest symmetric positive definite Toeplitz | ||
| %% matrix to a given symmetric matrix F. | ||
| %% | ||
| %% max -y0 | ||
| %% s.t. y0*B + T(y) (S>=) 0 | ||
| %% [y0; gam.*y] + [0; q./gam] (Q>=) 0 | ||
| %% | ||
| %% where B = diag([zeros(n,1); 1]) | ||
| %% q(1) = - Tr(F); q(k+1) = -sum of upper and lower kth diagonals of F | ||
| %% gam(1) = sqrt(n); gam(k) = sqrt(2*(n-k+1)) for k=2:n | ||
| %%************************************************************************* | ||
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| function [blk,At,C,b] = ToeplitzApproxSQQ(F) | ||
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| n = length(F); | ||
| gam = sqrt([n, 2*(n-1:-1:1)]'); | ||
| q = zeros(n,1); | ||
| q(1) = -sum(diag(F)); | ||
| for k=1:n-1 | ||
| q(k+1) = -2*sum(diag(F,k)); | ||
| end | ||
| beta = norm(F,'fro')^2; | ||
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| blk{1,1} = 's'; blk{1,2} = n+1; | ||
| blk{2,1} = 'q'; blk{2,2} = n+1; | ||
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| b = [-1; zeros(n,1)]; | ||
| C{1,1} = sparse(n+1,n+1); | ||
| C{2,1} = [0; q./gam]; | ||
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| Acell = cell(1,n+1); | ||
| Acell{1} = -spconvert([n+1,n+1,1]); | ||
| Acell{2} = -spdiags([ones(n,1); 0],0,n+1,n+1); | ||
| for k = 1:n-1 | ||
| tmp = -spdiags([ones(n,1); 0],k,n+1,n+1); | ||
| Acell{k+2} = tmp + tmp'; | ||
| end | ||
| At(1,1) = svec(blk(1,:),Acell,1); | ||
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| At{2,1} = -spdiags([1; gam],0,n+1,n+1); | ||
| %%************************************************************************* | ||
| %% ToeplitzApproxSQQ: find the nearest symmetric positive definite Toeplitz | ||
| %% matrix to a given symmetric matrix F. | ||
| %% | ||
| %% max -y0 | ||
| %% s.t. y0*B + T(y) (S>=) 0 | ||
| %% [y0; gam.*y] + [0; q./gam] (Q>=) 0 | ||
| %% | ||
| %% where B = diag([zeros(n,1); 1]) | ||
| %% q(1) = - Tr(F); q(k+1) = -sum of upper and lower kth diagonals of F | ||
| %% gam(1) = sqrt(n); gam(k) = sqrt(2*(n-k+1)) for k=2:n | ||
| %%***************************************************************** | ||
| %% SDPT3: version 4.0 | ||
| %% Copyright (c) 1997 by | ||
| %% Kim-Chuan Toh, Michael J. Todd, Reha H. Tutuncu | ||
| %% Last Modified: 16 Sep 2004 | ||
| %%***************************************************************** | ||
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| function [blk,At,C,b] = ToeplitzApproxSQQ(F) | ||
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| n = length(F); | ||
| gam = sqrt([n, 2*(n-1:-1:1)]'); | ||
| q = zeros(n,1); | ||
| q(1) = -sum(diag(F)); | ||
| for k=1:n-1 | ||
| q(k+1) = -2*sum(diag(F,k)); | ||
| end | ||
| beta = norm(F,'fro')^2; | ||
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| blk{1,1} = 's'; blk{1,2} = n+1; | ||
| blk{2,1} = 'q'; blk{2,2} = n+1; | ||
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| b = [-1; zeros(n,1)]; | ||
| C{1,1} = sparse(n+1,n+1); | ||
| C{2,1} = [0; q./gam]; | ||
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| Acell = cell(1,n+1); | ||
| Acell{1} = -spconvert([n+1,n+1,1]); | ||
| Acell{2} = -spdiags([ones(n,1); 0],0,n+1,n+1); | ||
| for k = 1:n-1 | ||
| tmp = -spdiags([ones(n,1); 0],k,n+1,n+1); | ||
| Acell{k+2} = tmp + tmp'; | ||
| end | ||
| At(1,1) = svec(blk(1,:),Acell,1); | ||
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| At{2,1} = -spdiags([1; gam],0,n+1,n+1); | ||
| %%*********************************************************************** |
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