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A problem with no strictly complementary solution

>> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
>> % Example from AHO1 where no strictly
>> % complementary solution exists
>> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
>> C = [0 0 0; 0 0 0; 0 0 1];
>> A1 = [1 0 0; 0 0 0; 0 0 0];
>> A2 = [0 0 1; 0 1 0; 1 0 0];
>> A3 = [0 1 0; 1 0 0; 0 0 1];
>> b = [1 0 0]';
>> blk = 3;
>> m = 3;
>> makeA
>> 
>> initvars
>> sdp

tau =   0.9990,     scalefac =        1


sdp: elapsed time               =   0.39385 seconds
sdp: elapsed cpu time           =   0.40000 seconds
sdp: flops                      =   5.72750e+04
sdp: Number of iterations       =   12
sdp: final value of X.Z         =   2.312e-09
sdp: final primal infeasibility =   2.220e-16
sdp: final dual infeasibility   =   5.752e-17
sdp: primal objective value     =   1.1559195968314993e-09
sdp: dual objective value       =  -1.1559195968284586e-09
>> 
>> [sort(blkeig(X,blk))  -sort(blkeig(-Z,blk))]

ans =

   1.2096e-11   1.0000e+00
   6.7641e-05   3.3820e-05
   1.0000e+00   1.2096e-11

>> 
>> % the true solutions X and Z each have rank 1,
>> % but observe how the eigenvalues of the computed
>> % solution are much less accurate than for problems with 
>> % strictly complementary solutions, and are not indicative
>> % of the true ranks.
>> 
>> primalcond(A,blk,X,1.0e-03);   % confirms that solution is primal NONdegenerate
primalcond =   1.000e+00
>> dualcond(A,blk,Z,1.0e-03);     % confirms that solution is dual NONdegenerate
dualcond =   1.414e+00
>> sdpcond(A,b,C,blk,X,y,Z);      % condition number is infinite, since SC failed

sdpcond: gap                      =   2.312e-09
sdpcond: primal infeasibility     =   2.220e-16
sdpcond: dual infeasibility       =   5.752e-17
sdpcond: cond estimate of 3x3 block matrix =   4.124e+04
>> % cond estimate is not large because eigenvalues which should
>> % be zero are not very small



Madhu Nayakkankuppam
Fri Mar 28 00:48:56 EST 1997