A Sample Trace Minimization Problem

In this problem we minimize $\mbox{trace}(Y^*AY)$, where $A$ is a $12 \times 12$ second difference operator in one dimension and $Y$ is $12 \times 4$.

Note: we do not recommend that this problem be solved this way, since, for constant $A$, it can be done faster with conventional eigenvalue solvers.

>> !cp examples/tracemin/*.m .
>> randn('state',0);
>> A = Kinetic(12);
>> parameters(A);
>> Y0 = guess(4);
>> [fn,Yn] = sg_min(Y0,'frcg','euclidean');
iter    grad            F(Y)              flops
0       2.249340e+00    4.339273e+00         3807
1       1.635681e+00    1.774835e+00       106770
2       6.506188e-01    1.160615e+00       196319
3       4.400435e-01    1.043566e+00       295232
4       3.460854e-01    9.601858e-01       386244
5       3.418816e-01    9.122465e-01       471214
6       2.591414e-01    8.543284e-01       560763
7       1.225067e-01    8.319959e-01       661414
8       5.660065e-02    8.280403e-01       785192
9       2.346852e-02    8.271969e-01       935587
10      9.258692e-03    8.270659e-01      1105819
11      4.723177e-03    8.270398e-01      1261468
12      3.282461e-03    8.270316e-01      1417625
13      1.984453e-03    8.270281e-01      1576710
14      9.808233e-04    8.270270e-01      1726651
15      4.339561e-04    8.270267e-01      1876592
16      1.990167e-04    8.270267e-01      2022569

17      6.563434e-05    8.270267e-01      2163632
18      3.165945e-05    8.270267e-01      2322727
19      2.015557e-05    8.270267e-01      2468720
20      1.261539e-05    8.270267e-01      2612825
21      7.123104e-06    8.270267e-01      2748702
22      2.552485e-06    8.270267e-01      2935969
23      9.679096e-07    8.270267e-01      3119232
24      3.168742e-07    8.270267e-01      3287067
25      2.062080e-07    8.270267e-01      3472366

Figure 9.3 shows the convergence curve for this run.

Figure 9.3: Trace minimization problem