OSQP Settings Sequence Data Frame

Data frame default_osqp_sequence

Fast and effective sequence of OSQP settings that should be suitable for accurately solving most time series balancing problems. It is the default value of tsbalancing() argument osqp_settings_df.

#>   max_iter    sigma  eps_abs  eps_rel eps_prim_inf eps_dual_inf polish scaling
#> 1    4,000 1.00e-09 1.00e-06 1.00e-06     1.00e-07     1.00e-07   TRUE       0
#> 2   10,000 1.00e-15 1.00e-12 1.00e-12     1.00e-13     1.00e-13   TRUE       0
#> 3   10,000 1.00e-15 1.00e-12 1.00e-12     1.00e-13     1.00e-13   TRUE       0
#> 4   10,000 2.22e-16 2.22e-16 2.22e-16     2.22e-16     2.22e-16   TRUE       0
#>   prior_scaling require_polished
#> 1          TRUE             TRUE
#> 2          TRUE             TRUE
#> 3         FALSE            FALSE
#> 4          TRUE            FALSE

Data frame alternate_osqp_sequence

Alternative slower sequence of OSQP settings that could help achieve more precision, when needed, especially when combined with argument full_sequence = TRUE.

#>    max_iter    sigma  eps_abs  eps_rel eps_prim_inf eps_dual_inf polish scaling
#> 1    10,000 1.00e-15 1.00e-12 1.00e-12     1.00e-13     1.00e-13   TRUE       0
#> 2    10,000 1.00e-15 1.00e-12 1.00e-12     1.00e-13     1.00e-13   TRUE      10
#> 3    10,000 1.00e-12 1.00e-09 1.00e-09     1.00e-10     1.00e-10   TRUE       0
#> 4    10,000 1.00e-12 1.00e-09 1.00e-09     1.00e-10     1.00e-10   TRUE      10
#> 5    10,000 1.00e-09 1.00e-06 1.00e-06     1.00e-07     1.00e-07   TRUE       0
#> 6    10,000 1.00e-09 1.00e-06 1.00e-06     1.00e-07     1.00e-07   TRUE      10
#> 7    10,000 1.00e-06 1.00e-03 1.00e-03     1.00e-04     1.00e-04   TRUE       0
#> 8    10,000 1.00e-06 1.00e-03 1.00e-03     1.00e-04     1.00e-04   TRUE      10
#> 9    10,000 2.22e-16 2.22e-16 2.22e-16     2.22e-16     2.22e-16   TRUE       0
#> 10   10,000 2.22e-16 2.22e-16 2.22e-16     2.22e-16     2.22e-16   TRUE      10
#> 11   10,000 1.00e-15 1.00e-12 1.00e-12     1.00e-13     1.00e-13   TRUE       0
#> 12   10,000 1.00e-12 1.00e-09 1.00e-09     1.00e-10     1.00e-10   TRUE       0
#> 13   10,000 1.00e-09 1.00e-06 1.00e-06     1.00e-07     1.00e-07   TRUE       0
#> 14   10,000 1.00e-06 1.00e-03 1.00e-03     1.00e-04     1.00e-04   TRUE       0
#> 15   10,000 2.22e-16 2.22e-16 2.22e-16     2.22e-16     2.22e-16   TRUE       0
#>    prior_scaling require_polished
#> 1          FALSE             TRUE
#> 2          FALSE             TRUE
#> 3          FALSE             TRUE
#> 4          FALSE             TRUE
#> 5          FALSE             TRUE
#> 6          FALSE             TRUE
#> 7          FALSE             TRUE
#> 8          FALSE             TRUE
#> 9          FALSE             TRUE
#> 10         FALSE             TRUE
#> 11          TRUE             TRUE
#> 12          TRUE             TRUE
#> 13          TRUE             TRUE
#> 14          TRUE             TRUE
#> 15          TRUE             TRUE

Details

With the exception of prior_scaling and require_polished, all columns of the data frame must correspond to a OSQP setting. Default OSQP values are used for any setting not specified in this data frame. Visit https://osqp.org/docs/interfaces/solver_settings.html for all available OSQP settings. Note that the OSQP verbose setting is actually controlled through tsbalancing() arguments quiet and display_level (i.e., column verbose in a OSQP settings sequence data frame would be ignored).

Each row of a OSQP settings sequence data frame represents one attempt at solving a balancing problem with the corresponding OSQP settings. The solving sequence stops as soon as a valid solution is obtained (a solution for which all constraint discrepancies are smaller or equal to the tolerance specified with tsbalancing() argument validation_tol) unless column require_polished = TRUE, in which case a polished solution from OSQP (status_polish = 1) would also be required to stop the sequence. Constraint discrepancies correspond to \(\mathrm{max}(0, l - Ax, Ax - u)\) with constraints defined as \(l \le Ax \le u\). In the event where a satisfactory solution cannot be obtained after having gone through the entire sequence, tsbalancing() returns the solution that generated the smallest total constraint discrepancies among valid solutions, if any, or among all solutions, otherwise. Note that running the entire solving sequence can be enforced by specifying tsbalancing() argument full_sequence = TRUE. Rows with column prior_scaling = TRUE have the problem data scaled prior to solving with OSQP, using the average of the free (nonbinding) problem values as the scaling factor.

In addition to specifying a custom-made OSQP settings sequence data frame with argument osqp_settings_df, one can also specify osqp_settings_df = NULL which would result in a single solving attempt with default OSQP values for all settings along with prior_scaling = FALSE and require_polished = FALSE. Note that it is recommended, however, to first try data frames default_osqp_sequence and alternate_osqp_sequence, along with full_sequence = TRUE if necessary, before considering other alternatives.

Recommended Approach

Start with the default tsbalancing() solving sequence (osqp_settings_df = default_osqp_sequence and full_sequence = FALSE). Then, if more precision is needed, try with:

1. full_sequence = TRUE
2. osqp_settings_df = alternate_osqp_sequence
3. osqp_settings_df = alternate_osqp_sequence and full_sequence = TRUE

In practice, specifying full_sequence = TRUE should be enough when more precision is needed (at the expense of execution time, obviously). Only in rare occasions should you need to use the alternate_osqp_sequence data frame, which will often be even more costly in terms of execution time especially when combined with full_sequence = TRUE.

Guiding Principles

The following is a summary of the lessons learned while developing tsbalancing() and experimenting with the OSQP solver. They are the guiding principles that lead to both OSQP settings sequence data frames presented earlier. Note that these observations apply to time series balancing problems as solved by tsbalancing() and may not directly apply to other types of quadratic problems.

• Data preconditioning options available in OSQP (with the scaling setting) are not sufficient for some (badly scaled) problems. External (prior) data scaling (prior_scaling = TRUE) is sometimes necessary for OSQP to converge at a decent pace and generate precise enough solutions in a reasonable number of iterations.
• Prior data scaling often reduces execution time (the required number of iterations to achieve the specified precision) and greatly increases the likelihood of polished solutions.
• Polished solutions are always very precise, even when prior data scaling is performed (i.e., the solution in the original scale will usually still be precise enough).
• While polished solutions from prior-scaled data are usually more precise than unpolished solutions from non prior-scaled data, the most precise solutions correspond to polished solutions from non prior-scaled data.
• Smaller sigma and tolerance (eps_*) settings result in more precise solutions but take longer to run (require more iterations).
• Enough precision is usually obtained after 10,000 iterations with small values for the sigma and tolerance (eps_*) settings.
• The default OSQP values for alpha and the various settings associated to \(\rho\) (*rho*) are sufficient (they work well). Reducing the sigma and tolerance (eps_*) settings and performing prior data scaling is sufficient to obtain precise solutions in a reasonable number of iterations.
• Keeping the same scale between the sigma and tolerance (eps_*) settings as the default OSQP values works well and is used in both OSQP settings sequence data frames, i.e.:
  • eps_abs = eps_rel = 1,000 * sigma
  • eps_prim_inf = eps_dual_inf = 100 * sigma
  • (and consequently) eps_abs = eps_rel = 10 * eps_prim_inf = 10 * eps_dual_inf
• The machine epsilon (.Machine$double.eps) for the sigma and tolerance (eps_*) settings, which basically forces the maximum number of iterations, is used as a last resort in both OSQP settings sequence data frames.

Summary - default sequence (data frame default_osqp_sequence)

• Geared towards achieving both fast and precise solutions.
• First try to get (fast) polished solutions on prior-scaled data before attempting solving the original (non-scaled) problem data.
• Make a final attempt on prior-scaled data with the machine epsilon for the sigma and tolerance (eps_*) settings.

Summary - alternative sequence (data frame alternate_osqp_sequence)

• Geared towards achieving precise solutions at the expense of execution time.
• Somewhat similar to a brute force or try all approach (especially when combined with full_sequence = TRUE).
• Small sigma and tolerance (eps_*) settings that gradually increase, with the machine epsilon as a last attempt.
• Polished solutions are required for every step of the sequence (the best unpolished solution is returned if no polished solution could be obtained at the end of the entire sequence).
• Maximum of 10,000 iterations for every step of the sequence
• First try to get polished solutions from non prior-scaled data (the most precise solutions), then try with prior-scaled data.