Update trust_region_step_exact_qr

[YigitElma]

• Updates the initial guess for $\alpha$ to be 0 instead of 0.001* $\alpha$ _upper. I found out that the solution of $\alpha$ is usually pretty small. This reduces the total amount of QR's in the while loop.
• Removes the last QR outside the while loop of trust_region_step_exact_qr. The loop condition already satisfies the step norm to be around trust radius.

Misc

• Adds execute_on_cpu flag to desc.examples.get

[github-actions]

|             benchmark_name             |         dt(%)          |         dt(s)          |        t_new(s)        |        t_old(s)        | 
| -------------------------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- |
 test_build_transform_fft_midres         |     +3.04 +/- 6.85     | +1.86e-02 +/- 4.19e-02 |  6.30e-01 +/- 3.2e-02  |  6.12e-01 +/- 2.7e-02  |
 test_build_transform_fft_highres        |     -0.39 +/- 4.57     | -3.86e-03 +/- 4.48e-02 |  9.77e-01 +/- 3.7e-02  |  9.81e-01 +/- 2.5e-02  |
 test_equilibrium_init_lowres            |     -2.36 +/- 5.24     | -9.64e-02 +/- 2.14e-01 |  3.98e+00 +/- 1.3e-01  |  4.08e+00 +/- 1.7e-01  |
 test_objective_compile_atf              |     -2.47 +/- 3.60     | -2.01e-01 +/- 2.93e-01 |  7.94e+00 +/- 1.6e-01  |  8.14e+00 +/- 2.5e-01  |
 test_objective_compute_atf              |     +2.54 +/- 4.85     | +2.68e-04 +/- 5.11e-04 |  1.08e-02 +/- 4.2e-04  |  1.05e-02 +/- 2.9e-04  |
 test_objective_jac_atf                  |     +0.93 +/- 2.62     | +1.82e-02 +/- 5.16e-02 |  1.99e+00 +/- 3.2e-02  |  1.97e+00 +/- 4.1e-02  |
 test_perturb_1                          |     -0.09 +/- 2.65     | -1.31e-02 +/- 3.87e-01 |  1.46e+01 +/- 2.6e-01  |  1.46e+01 +/- 2.9e-01  |
 test_proximal_jac_atf                   |     -0.72 +/- 1.74     | -5.93e-02 +/- 1.43e-01 |  8.15e+00 +/- 7.0e-02  |  8.21e+00 +/- 1.3e-01  |
 test_proximal_freeb_compute             |     -2.74 +/- 1.46     | -5.49e-03 +/- 2.92e-03 |  1.95e-01 +/- 9.8e-04  |  2.00e-01 +/- 2.8e-03  |
 test_build_transform_fft_lowres         |     +5.08 +/- 3.92     | +2.69e-02 +/- 2.07e-02 |  5.56e-01 +/- 1.6e-02  |  5.29e-01 +/- 1.3e-02  |
 test_equilibrium_init_medres            |     -0.80 +/- 4.40     | -3.39e-02 +/- 1.86e-01 |  4.19e+00 +/- 1.1e-01  |  4.22e+00 +/- 1.5e-01  |
 test_equilibrium_init_highres           |     -1.41 +/- 3.35     | -7.95e-02 +/- 1.89e-01 |  5.57e+00 +/- 1.2e-01  |  5.65e+00 +/- 1.5e-01  |
 test_objective_compile_dshape_current   |     +1.17 +/- 3.08     | +4.57e-02 +/- 1.20e-01 |  3.94e+00 +/- 1.1e-01  |  3.89e+00 +/- 4.3e-02  |
 test_objective_compute_dshape_current   |     -1.12 +/- 1.45     | -4.08e-05 +/- 5.28e-05 |  3.60e-03 +/- 2.9e-05  |  3.64e-03 +/- 4.4e-05  |
 test_objective_jac_dshape_current       |     +0.34 +/- 11.75    | +1.44e-04 +/- 4.91e-03 |  4.19e-02 +/- 4.5e-03  |  4.18e-02 +/- 1.9e-03  |
 test_perturb_2                          |     +0.55 +/- 1.83     | +1.05e-01 +/- 3.48e-01 |  1.92e+01 +/- 3.3e-01  |  1.91e+01 +/- 1.1e-01  |
 test_proximal_freeb_jac                 |     +0.85 +/- 1.74     | +6.43e-02 +/- 1.31e-01 |  7.63e+00 +/- 1.1e-01  |  7.57e+00 +/- 6.4e-02  |
 test_solve_fixed_iter                   |    -23.20 +/- 68.55    | -1.18e+00 +/- 3.49e+00 |  3.91e+00 +/- 2.7e+00  |  5.09e+00 +/- 2.2e+00  |

[f0uriest]

isn't this the same as the cho option but using LU instead of cholesky?

[YigitElma]

isn't this the same as the cho option but using LU instead of cholesky?

Yes, pretty similar. jax.scipy.linlag.solve with symmetric A should call LAPACK ?sysv (docs), and it is something like LU. For finding alpha, I used Secant method.

Cholesky is also really fast (maybe faster than this one), but maybe this is more accurate, that's why I wanted to test it. If I remember correctly the problem with Cholesky was the accuracy.

[f0uriest]

Generally the innaccuracy comes from explitly forming J^T J when J is poorly conditioned, so I don't think you'll see any major improvements with this. Also cholesky should be faster.

[dpanici]

run on multiple threads

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 95.56%. Comparing base (2741269) to head (3c3dadc).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1363      +/-   ##
==========================================
- Coverage   95.57%   95.56%   -0.01%     
==========================================
  Files          96       96              
  Lines       24405    24406       +1     
==========================================
- Hits        23325    23324       -1     
- Misses       1080     1082       +2     

Files with missing lines	Coverage Δ
desc/examples/__init__.py	100.00% <100.00%> (ø)
desc/optimize/tr_subproblems.py	99.44% <100.00%> (-0.01%)	⬇️

... and 1 file with indirect coverage changes