tol_rel_obj doesn't seem to work in optimizing()

Summary:

tol_rel_obj not terminating when I think it should.

Description:

Unless I set tol_rel_obj to very high values, it does not cause convergence.

Reproducible Steps:

Should be evident in any optimization model. Here's what I get:

> model <- stan_model(...)
> fit_init <- optimizing(model, data = stan_data_tasics,
                       init = 0.1,
		       tol_rel_obj = 1e-1, tol_obj = 0.01e7,
		       tol_rel_grad = 1e1, tol_grad = 1e1,
                       tol_param = 1e-2,
		       iter = 1000,
		       history_size = 10,
                       as_vector = FALSE,
                       verbose = TRUE)
...
Chain 1:       89  -4.86797e+07       11.8801        105693      0.5518      0.5518      164   
Chain 1:     Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes 
Chain 1:       99  -4.66481e+07       21.5455        110251           1           1      174   
Chain 1:     Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes 
Chain 1:      104  -4.58516e+07       4.36499         75484      0.8113      0.8113      179   
Chain 1: Optimization terminated normally: 
Chain 1:   Convergence detected: absolute change in objective function was below tolerance

So it seems like tol_obj is working as expected.

But if I set tol_obj so that doesn't cause convergence, then I don't get termination until it's well beyond the relative tolerance I set:

fit_init <- optimizing(model, data = stan_data_tasics,
                       init = 0.1,
		       tol_rel_obj = 1e10, tol_obj = 1e-10,
		       tol_rel_grad = 1e1, tol_grad = 1e1,
                       tol_param = 1e-2,
		       iter = 1000,
		       history_size = 10,
                       as_vector = FALSE,
                       verbose = TRUE)
...
Chain 1:      559  -3.72821e+07       2.68462       5413.19           1           1      678   
Chain 1:     Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes 
Chain 1:      569  -3.72777e+07      0.922598       5027.67      0.7192      0.7192      688   
Chain 1:     Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes 
Chain 1:      579  -3.72737e+07      0.177795       3061.21           1           1      699   
Chain 1:     Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes 
Chain 1:      589  -3.72697e+07      0.155879       3135.34      0.8214      0.8214      710   
Chain 1:     Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes 
Chain 1:      599  -3.72663e+07      0.884956       3098.07      0.8752      0.8752      720   
Chain 1:     Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes 
Chain 1:      602  -3.72657e+07       2.57617       9432.57           1           1      723   
Chain 1: Optimization terminated normally: 
Chain 1:   Convergence detected: relative change in objective function was below tolerance

If I set tol_rel_obj really high, like to 1e100, it terminates very fast.

Apologies in advance if this is just be some rolling average calculation or something I don't understand.

As an aside, it would help immensely if the as_vector = FALSE option in the doc explained that you get the results in structured form.

sessionInfo()

> sessionInfo()
R version 4.0.1 (2020-06-06)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Catalina 10.15.7

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] tools     stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] rstan_2.19.3         StanHeaders_2.21.0-5 ggpubr_0.4.0         ggplot2_3.3.2       

loaded via a namespace (and not attached):
 [1] tidyselect_1.1.0   purrr_0.3.4        haven_2.3.1        lattice_0.20-41    carData_3.0-4      colorspace_1.4-1   vctrs_0.3.1       
 [8] generics_0.0.2     stats4_4.0.1       loo_2.2.0          rlang_0.4.6        pkgbuild_1.0.8     pillar_1.4.4       foreign_0.8-80    
[15] glue_1.4.1         withr_2.2.0        readxl_1.3.1       matrixStats_0.56.0 lifecycle_0.2.0    munsell_0.5.0      ggsignif_0.6.0    
[22] gtable_0.3.0       cellranger_1.1.0   zip_2.0.4          codetools_0.2-16   labeling_0.3       inline_0.3.15      rio_0.5.16        
[29] forcats_0.5.0      callr_3.4.3        ps_1.3.3           parallel_4.0.1     curl_4.3           fansi_0.4.1        broom_0.5.6       
[36] Rcpp_1.0.4.6       backports_1.1.8    scales_1.1.1       RcppParallel_5.0.2 abind_1.4-5        farver_2.0.3       gridExtra_2.3     
[43] digest_0.6.25      hms_0.5.3          stringi_1.4.6      openxlsx_4.1.5     rstatix_0.6.0      processx_3.4.2     dplyr_1.0.0       
[50] grid_4.0.1         cli_2.0.2          magrittr_1.5       tibble_3.0.1       crayon_1.3.4       car_3.0-9          tidyr_1.1.0       
[57] pkgconfig_2.0.3    ellipsis_0.3.1     data.table_1.13.0  prettyunits_1.1.1  assertthat_0.2.1   R6_2.4.1           nlme_3.1-148      
[64] compiler_4.0.1    

Start by reproducing the two optimizing() calls in the issue and then trace the optimizing() implementation where tol_rel_obj and tol_obj are handled. Done means explaining whether the relative tolerance is evaluated as intended and documenting or correcting the behavior, with the as_vector documentation aside addressed if it belongs to this issue.