JSO Compliance: R2 #139

MaxenceGollier · 2024-06-08T14:54:50Z

Rewrote R2_alg.jl to be more in phase with other JSO solvers.

Added a function SolverCore.solve!( R2Solver, RegularizedNLPModel, stats ) which implements the algorithm.
Callbacks are now available
Changed implementations of the previous R2 functions to fit this new approach but made sure to keep the previous signatures and returns so that nothing else in this package is impacted. This makes the code messy in some places and should be removed. (for example, there is no need for Fhist, Hhist, etc. in my opinion). Also, the documentation i added does not show these functions.
On a small benchmark, there seems to be less allocations with this code.

I initially just wanted to add callbacks so i don't know if this is relevant. What do you think @dpo, @geoffroyleconte, @MohamedLaghdafHABIBOULLAH ?

dpo · 2024-06-08T18:47:59Z

Thank you, @MaxenceGollier! We'll look in detail.

Yes, callbacks are necessary. See #131.

ps: you can mark functions that should eventually be removed with Base.@deprecate (https://docs.julialang.org/en/v1/base/base/#Base.@deprecate).

dpo

Just a few quick simple comments for now.

src/R2_alg.jl

dpo

Thank you. Here are a few more comments.

I would like to see a comparison of the old and new R2 on the problems we have to make sure the results are the same (number of iterations and final values (f, h, ξ, …).

src/R2_alg.jl

MaxenceGollier · 2024-06-10T14:43:27Z

First, this version:

julia> using LinearAlgebra, NLPModels, NLPModels, NLPModelsModifiers, RegularizedProblems, ProximalOperators, Random
julia> Random.seed!(0)
julia> bpdn = bpdn_model(1)[1]
julia> λ = norm(grad(bpdn, zeros(bpdn.meta.nvar)), Inf) / 10
julia> h = NormL1(λ)

julia> import Pkg; Pkg.add(url = "https://github.com/MaxenceGollier/RegularizedOptimization.jl.git", rev = "RegularizedModels")
julia> using RegularizedOptimization
julia> options = ROSolverOptions(ν = 1.0, β = 1e16, ϵa = 1e-6, ϵr = 1e-6, verbose = 1)

julia> R2(bpdn, h, options)
[ Info:   iter     f(x)     h(x)   √(ξ/ν)        ρ        σ      ‖x‖      ‖s‖
[ Info:      0  1.8e+00  0.0e+00  1.3e+00  5.5e-01  1.0e+00  0.0e+00  1.3e+00               =
[ Info:      1  2.5e-01  5.8e-01  3.3e-01  9.5e-01  1.0e+00  1.3e+00  3.2e-01               ↘
[ Info:      2  2.1e-01  5.2e-01  2.5e-01  8.5e-01  3.3e-01  1.5e+00  7.3e-01               =
[ Info:      3  1.1e-01  4.5e-01  1.8e-01  8.6e-01  3.3e-01  2.0e+00  5.3e-01               =
[ Info:      4  1.3e-01  3.5e-01  1.5e-01  6.5e-01  3.3e-01  2.2e+00  4.4e-01               =
[ Info:      5  5.0e-02  3.9e-01  6.1e-02  7.1e-01  3.3e-01  2.6e+00  1.4e-01               =
[ Info:      6  4.9e-02  3.8e-01  2.8e-02  3.4e-01  3.3e-01  2.7e+00  8.3e-02               =
[ Info:      7  3.9e-02  3.9e-01  1.4e-02  2.0e-01  3.3e-01  2.7e+00  4.1e-02               =
[ Info:      8  4.1e-02  3.9e-01  8.9e-03  1.8e-01  3.3e-01  2.7e+00  2.7e-02               =
[ Info:      9  3.9e-02  3.9e-01  5.8e-03  1.7e-01  3.3e-01  2.7e+00  1.7e-02               =
[ Info:     10  4.0e-02  3.9e-01  3.8e-03  1.7e-01  3.3e-01  2.7e+00  1.2e-02               =
[ Info:     11  4.0e-02  3.9e-01  2.5e-03  1.7e-01  3.3e-01  2.7e+00  7.6e-03               =
[ Info:     12  4.0e-02  3.9e-01  1.7e-03  1.7e-01  3.3e-01  2.7e+00  5.0e-03               =
[ Info:     13  4.0e-02  3.9e-01  1.1e-03  1.7e-01  3.3e-01  2.7e+00  3.3e-03               =
[ Info:     14  4.0e-02  3.9e-01  7.4e-04  1.7e-01  3.3e-01  2.7e+00  2.2e-03               =
[ Info:     15  4.0e-02  3.9e-01  4.9e-04  1.7e-01  3.3e-01  2.7e+00  1.5e-03               =
[ Info:     16  4.0e-02  3.9e-01  3.2e-04  1.7e-01  3.3e-01  2.7e+00  9.7e-04               =
[ Info:     17  4.0e-02  3.9e-01  2.1e-04  1.7e-01  3.3e-01  2.7e+00  6.4e-04               =
[ Info:     18  4.0e-02  3.9e-01  1.4e-04  1.7e-01  3.3e-01  2.7e+00  4.2e-04               =
[ Info:     19  4.0e-02  3.9e-01  9.3e-05  1.7e-01  3.3e-01  2.7e+00  2.8e-04               =
[ Info:     20  4.0e-02  3.9e-01  6.2e-05  1.7e-01  3.3e-01  2.7e+00  1.8e-04               =
[ Info:     21  4.0e-02  3.9e-01  4.1e-05  1.7e-01  3.3e-01  2.7e+00  1.2e-04               =
[ Info:     22  4.0e-02  3.9e-01  2.7e-05  1.7e-01  3.3e-01  2.7e+00  8.1e-05               =
[ Info:     23  4.0e-02  3.9e-01  1.8e-05  1.7e-01  3.3e-01  2.7e+00  5.4e-05               =
[ Info:     24  4.0e-02  3.9e-01  1.2e-05  1.7e-01  3.3e-01  2.7e+00  3.5e-05               =
[ Info:     25  4.0e-02  3.9e-01  7.8e-06  1.7e-01  3.3e-01  2.7e+00  2.3e-05               =
[ Info:     26  4.0e-02  3.9e-01  5.2e-06  1.7e-01  3.3e-01  2.7e+00  1.5e-05               =
[ Info:     27  4.0e-02  3.9e-01  3.4e-06  1.7e-01  3.3e-01  2.7e+00  1.0e-05               =
[ Info: R2: terminating with √(ξ/ν) = 2.2587837268169823e-6
"Execution stats: first-order stationary"

Restarting Julia, the old version gives

julia> using LinearAlgebra, NLPModels, NLPModels, NLPModelsModifiers, RegularizedProblems, ProximalOperators, Random
julia> Random.seed!(0)
julia> bpdn = bpdn_model(1)[1]
julia> λ = norm(grad(bpdn, zeros(bpdn.meta.nvar)), Inf) / 10
julia> h = NormL1(λ)

julia> import Pkg; Pkg.add(url = "https://github.com/JuliaSmoothOptimizers/RegularizedOptimization.jl.git")
julia> using RegularizedOptimization
julia> options = ROSolverOptions(ν = 1.0, β = 1e16, ϵa = 1e-6, ϵr = 1e-6, verbose = 10)

julia> R2(bpdn, h, options)
[ Info:   iter     f(x)     h(x)  √(ξ/ν)        ρ       σ     ‖x‖     ‖s‖
[ Info:      1  1.8e+00  0.0e+00 1.3e+00  5.5e-01 1.0e+00 0.0e+00 1.3e+00 =
[ Info:      2  2.5e-01  5.8e-01 3.3e-01  9.5e-01 1.0e+00 1.3e+00 3.2e-01 ↘
[ Info:      3  2.1e-01  5.2e-01 2.5e-01  8.5e-01 3.3e-01 1.5e+00 7.3e-01 =
[ Info:      4  1.1e-01  4.5e-01 1.8e-01  8.6e-01 3.3e-01 2.0e+00 5.3e-01 =
[ Info:      5  1.3e-01  3.5e-01 1.5e-01  6.5e-01 3.3e-01 2.2e+00 4.4e-01 =
[ Info:      6  5.0e-02  3.9e-01 6.1e-02  7.1e-01 3.3e-01 2.6e+00 1.4e-01 =
[ Info:      7  4.9e-02  3.8e-01 2.8e-02  3.4e-01 3.3e-01 2.7e+00 8.3e-02 =
[ Info:      8  3.9e-02  3.9e-01 1.4e-02  2.0e-01 3.3e-01 2.7e+00 4.1e-02 =
[ Info:      9  4.1e-02  3.9e-01 8.9e-03  1.8e-01 3.3e-01 2.7e+00 2.7e-02 =
[ Info:     10  3.9e-02  3.9e-01 5.8e-03  1.7e-01 3.3e-01 2.7e+00 1.7e-02 =
[ Info:     11  4.0e-02  3.9e-01 3.8e-03  1.7e-01 3.3e-01 2.7e+00 1.2e-02 =
[ Info:     12  4.0e-02  3.9e-01 2.5e-03  1.7e-01 3.3e-01 2.7e+00 7.6e-03 =
[ Info:     13  4.0e-02  3.9e-01 1.7e-03  1.7e-01 3.3e-01 2.7e+00 5.0e-03 =
[ Info:     14  4.0e-02  3.9e-01 1.1e-03  1.7e-01 3.3e-01 2.7e+00 3.3e-03 =
[ Info:     15  4.0e-02  3.9e-01 7.4e-04  1.7e-01 3.3e-01 2.7e+00 2.2e-03 =
[ Info:     16  4.0e-02  3.9e-01 4.9e-04  1.7e-01 3.3e-01 2.7e+00 1.5e-03 =
[ Info:     17  4.0e-02  3.9e-01 3.2e-04  1.7e-01 3.3e-01 2.7e+00 9.7e-04 =
[ Info:     18  4.0e-02  3.9e-01 2.1e-04  1.7e-01 3.3e-01 2.7e+00 6.4e-04 =
[ Info:     19  4.0e-02  3.9e-01 1.4e-04  1.7e-01 3.3e-01 2.7e+00 4.2e-04 =
[ Info:     20  4.0e-02  3.9e-01 9.3e-05  1.7e-01 3.3e-01 2.7e+00 2.8e-04 =
[ Info:     21  4.0e-02  3.9e-01 6.2e-05  1.7e-01 3.3e-01 2.7e+00 1.8e-04 =
[ Info:     22  4.0e-02  3.9e-01 4.1e-05  1.7e-01 3.3e-01 2.7e+00 1.2e-04 =
[ Info:     23  4.0e-02  3.9e-01 2.7e-05  1.7e-01 3.3e-01 2.7e+00 8.1e-05 =
[ Info:     24  4.0e-02  3.9e-01 1.8e-05  1.7e-01 3.3e-01 2.7e+00 5.4e-05 =
[ Info:     25  4.0e-02  3.9e-01 1.2e-05  1.7e-01 3.3e-01 2.7e+00 3.5e-05 =
[ Info:     26  4.0e-02  3.9e-01 7.8e-06  1.7e-01 3.3e-01 2.7e+00 2.3e-05 =
[ Info:     27  4.0e-02  3.9e-01 5.2e-06  1.7e-01 3.3e-01 2.7e+00 1.5e-05 =
[ Info:     28  4.0e-02  3.9e-01 3.4e-06  1.7e-01 3.3e-01 2.7e+00 1.0e-05 =
[ Info:     29  4.0e-02  3.9e-01 2.3e-06          3.3e-01 2.7e+00 6.8e-06
[ Info: R2: terminating with √(ξ/ν) = 2.2587837268169823e-6
"Execution stats: first-order stationary"

There is just a small difference in the fact that the iterates start at 0 and don't print the last one, I think the last one is useless because except for the stationarity measure nothing changes.

MaxenceGollier · 2024-06-10T14:56:11Z

Then, for benchmarks,

For this version:

julia> using LinearAlgebra, NLPModels, NLPModels, NLPModelsModifiers, RegularizedProblems, ProximalOperators, Random, BenchmarkTools
julia> Random.seed!(0)
julia> bpdn = bpdn_model(1)[1]
julia> λ = norm(grad(bpdn, zeros(bpdn.meta.nvar)), Inf) / 10
julia> h = NormL1(λ)

julia> import Pkg; Pkg.add(url = "https://github.com/MaxenceGollier/RegularizedOptimization.jl.git", rev = "RegularizedModels")
julia> using RegularizedOptimization
julia> options = ROSolverOptions(ν = 1.0, β = 1e16, ϵa = 1e-6, ϵr = 1e-6, verbose = 0)

julia>  @benchmark R2(bpdn, h, options)
BenchmarkTools.Trial: 6852 samples with 1 evaluation.
 Range (min … max):  345.100 μs … 404.146 ms  ┊ GC (min … max):  0.00% … 99.76%
 Time  (median):     558.700 μs               ┊ GC (median):     0.00%
 Time  (mean ± σ):   720.358 μs ±   4.890 ms  ┊ GC (mean ± σ):  11.14% ±  6.57%

          ▃▅█▇▆▄▃▂▁    ▂▄▃▂▁  ▁▂▂▂▁▂▁▁                          ▂
  ▇▃▁▅▃▅▅▁███████████████████████████████▇▇█▇▇▇▇▆▅▆▇▆▇▇▆▅▅▅▃▅▁▅ █
  345 μs        Histogram: log(frequency) by time       1.49 ms <

 Memory estimate: 407.41 KiB, allocs estimate: 7756.

and for the old version:

julia> using LinearAlgebra, NLPModels, NLPModels, NLPModelsModifiers, RegularizedProblems, ProximalOperators, Random, BenchmarkTools
julia> Random.seed!(0)
julia> bpdn = bpdn_model(1)[1]
julia> λ = norm(grad(bpdn, zeros(bpdn.meta.nvar)), Inf) / 10
julia> h = NormL1(λ)

julia> import Pkg; Pkg.add(url = "https://github.com/JuliaSmoothOptimizers/RegularizedOptimization.jl.git")
julia> using RegularizedOptimization
julia> options = ROSolverOptions(ν = 1.0, β = 1e16, ϵa = 1e-6, ϵr = 1e-6, verbose = 0)

julia> @benchmark R2(bpdn, h, options)
BenchmarkTools.Trial: 276 samples with 1 evaluation.
 Range (min … max):  13.377 ms … 457.888 ms  ┊ GC (min … max): 0.00% … 96.51%
 Time  (median):     15.351 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   18.084 ms ±  26.749 ms  ┊ GC (mean ± σ):  9.85% ±  7.25%

   █▆▆▄▁▂▂ ▁      ▃ ▂▂ ▂▄▄▃
  ▆█████████▆▆▅▆▆███████████▁▅▆▅▆▅▁▅▆▅▅▁▆▅▆▁▅▆▁▁▁▁▁▁▅▁▁▁▁▁▁▅▆▅ ▆
  13.4 ms       Histogram: log(frequency) by time      27.2 ms <

 Memory estimate: 3.68 MiB, allocs estimate: 222912.

MaxenceGollier · 2024-06-10T15:39:33Z

Changing the line @benchmark R2(bpdn, h, options) to @benchmark R2($bpdn, $h, $options) don't change anything.

dpo · 2024-06-11T15:10:25Z

@MaxenceGollier I fixed the demos. Could you please rebase your branch? That way, you'll have the latest fixes (builds on FreeBSD + the demos working again). The procedure goes like this:

> git checkout master

If origin in your clone points to this repository (you can find out with git remote -v), do

> git pull origin  master

If origin points to your fork instead, first do

> git remote add upstream https://github.com/JuliaSmoothOptimizers/RegularizedOptimization.jl.git
> git pull upstream master

Now head back to your branch and rebase

> git checkout RegularizedModels
> git rebase master

Hopefully, there are no conflicts to fix.

Co-authored-by: Dominique <[email protected]>

MaxenceGollier · 2024-06-12T10:59:29Z

There are still rests of the older version which are just kept to not destroy other implementations in this package that rely on the older version.
For example, I think the constructor

function R2Solver(
  x0::S,
  options::ROSolverOptions,
  l_bound::S,
  u_bound::S;
  ψ = nothing
)

is useless. Also, I don't see why we should use ROSolverOptions instead of keyword arguments.
Finally, storing

Fobj_hist::Vector{R}
Hobj_hist::Vector{R}
Complex_hist::Vector{Int}

in R2Solver is expensive and useless. Should we just keep it this way for the moment and maybe try to remove it later ?

dpo · 2024-06-12T21:54:35Z

Your branch still has conflicts.

Let’s keep “old” features for now. We’ll remove them later. I opened a PR a while ago to remove Fhist, etc., We can update it after this one’s been merged.

MaxenceGollier · 2024-06-13T11:53:38Z

Tests should pass now with 754feea

github-actions · 2024-06-15T22:37:06Z

Here are the
demos-results

dpo

Just a couple of minor comments.

src/R2_alg.jl

github-actions · 2024-06-19T15:21:06Z

Here are the
demos-results

Co-authored-by: Dominique <[email protected]>

codecov · 2024-06-22T12:41:33Z

Codecov Report

Attention: Patch coverage is 90.32258% with 12 lines in your changes missing coverage. Please review.

Project coverage is 60.21%. Comparing base (e0f214d) to head (6dfcb53).
Report is 1 commits behind head on master.

❗ Current head 6dfcb53 differs from pull request most recent head f0a273c

Please upload reports for the commit f0a273c to get more accurate results.

Files	Patch %	Lines
src/R2_alg.jl	90.32%	12 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #139      +/-   ##
==========================================
- Coverage   61.53%   60.21%   -1.32%     
==========================================
  Files          11       11              
  Lines        1292     1302      +10     
==========================================
- Hits          795      784      -11     
- Misses        497      518      +21

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src/R2_alg.jl

github-actions · 2024-06-22T14:33:50Z

Here are the
demos-results

Co-authored-by: Dominique <[email protected]>

github-actions · 2024-06-22T18:20:16Z

Here are the
demos-results

dpo · 2024-06-22T20:42:28Z

There are lots of large differences in the demos, in particular on the plots showing the number of inner iterations. I wonder if other solvers are able to call the new R2 and if the number of iterations is reported properly. Could you please check?

Among others, these tests show differences:

bpdn-inner-outer-constr-lm-r2-l0-linf
bpdn-inner-outer-constr-lm-r2-l1-linf
bpdn-inner-outer-constr-lmtr-r2-l0-linf
bpdn-inner-outer-constr-lmtr-r2-l1-linf
bpdn-inner-outer-r2-l1-l2 (this is R2 alone; the number of iterations goes from 20 to 35)
bpdn-inner-outer-r2-l1-linf (R2 alone)
and others.

Those problems are generated randomly but we initialize the seed, so they should be the same.

MaxenceGollier · 2024-06-23T11:28:48Z

Yes they do, when i test locally, all inner iterations are reported correctly. Can you show me with respect to what you are comparing ? I just see the demos-result for this branch (and not for the base one ?)

For example for bpdn-inner-outer-r2-l1-l2, I see less than 35 outer iterations on the plot so we do have between 20 and 35 iterations.

dpo · 2024-06-23T12:25:03Z

I was comparing to the demos in this PR: #138
There have been no changes to the solvers since that PR.
But you could easily regenerate the demo results locally from the main branch.

MaxenceGollier · 2024-06-23T13:43:44Z

It is weird because my local branch is up to date with this and when I run this locally on bpdn-inner-outer-constr-lm-r2-l0-linf, I get

┌ Info:  using LM to solve with
└   h = NormL0{Float64}(0.04974454739546478)
[ Info:  outer    inner     f(x)     h(x)     √ξ1      √ξ        ρ       σ     ‖x‖     ‖s‖   ‖Jₖ‖² reg
[ Info:      1        2  2.1e+00  0.0e+00 4.9e-01 3.2e-01  2.8e+00 1.0e+00 0.0e+00 6.0e-01 2.0e+00 ↘
[ Info:      2        7  1.6e+00  2.0e-01 7.5e-01 7.0e-01  1.6e+00 3.3e-01 6.0e-01 1.3e+00 1.3e+00 ↘
[ Info:      3        8  6.0e-01  4.0e-01 5.5e-01 6.0e-01  1.3e+00 1.1e-01 1.8e+00 1.4e+00 1.1e+00 ↘
[ Info:      4        9  4.4e-02  5.0e-01 1.6e-01 1.8e-01  1.1e+00 3.7e-02 2.8e+00 3.9e-01 1.0e+00 ↘
[ Info:      5       16  9.1e-03  5.0e-01 1.7e-02 2.0e-02  1.0e+00 1.2e-02 3.1e+00 4.6e-02 1.0e+00 ↘
[ Info:      6       15  8.7e-03  5.0e-01 5.9e-04 7.1e-04  1.0e+00 4.1e-03 3.1e+00 1.7e-03 1.0e+00 ↘
[ Info:      7        5  8.7e-03  5.0e-01 8.1e-06 9.7e-06  1.0e+00 1.4e-03 3.1e+00 2.1e-05 1.0e+00 ↘
[ Info:      8        1  8.7e-03  5.0e-01 1.1e-06 1.1e-06          4.6e-04 3.1e+00 1.1e-06 1.0e+00
[ Info: LM: terminating with √ξ1 = 1.069826582248765e-6

So I should have the same results as on #138, I don't really see where these plots are generated ? The stats.solver_specific[:SubsolverCounter] contains the correct values as well...

dpo · 2024-06-23T15:05:34Z

They’re generated by the scripts in the examples folder.

MaxenceGollier · 2024-06-26T12:05:04Z

The issue should be solved with f0a273c now.

MaxenceGollier · 2024-07-03T13:47:57Z

src/R2_alg.jl

-* `Fobj_hist`: an array with the history of values of the smooth objective
-* `Hobj_hist`: an array with the history of values of the nonsmooth objective
-* `Complex_hist`: an array with the history of number of inner iterations.
+ψ(s; xₖ) is either h(xₖ + s) or an approximation of h(xₖ + s), ‖⋅‖ is a user-defined norm and σₖ > 0 is the regularization parameter.


Is $||\cdot||$ really a user-defined norm ? Isn't it always the $\ell_2$ norm ?

Suggested change

ψ(s; xₖ) is either h(xₖ + s) or an approximation of h(xₖ + s), ‖⋅‖ is a user-defined norm and σₖ > 0 is the regularization parameter.

ψ(s; xₖ) is either h(xₖ + s) or an approximation of h(xₖ + s), ‖⋅‖ is a user-defined norm and σₖ > 0 is the regularization parameter.

Oui en effet, c'est la norme 2.

dpo requested changes Jun 8, 2024

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dpo reviewed Jun 10, 2024

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MaxenceGollier and others added 14 commits June 12, 2024 10:27

Add RegularizedNLPModels

c386150

Add RegularizedNLPModels to R2

6ed13e6

Update documentation

435838a

Change R2_alg with SolverCore.solve!

9e65f82

Fix bound issue, all tests pass

5895a85

Update documentation

9c5217b

Minor bug fixes

3ee28f6

Spacing and documentation update

c0d1905

Update documentation

86d2271

Update src/R2_alg.jl

61d7d20

Co-authored-by: Dominique <[email protected]>

Update src/R2_alg.jl

47247e7

Co-authored-by: Dominique <[email protected]>

Update documentation

4faf4ed

Update @info log_row position

dbc1189

Update @info log_row position

ebf1457

MaxenceGollier added 3 commits June 15, 2024 12:20

Update @info log_row position

ef3de43

Update @info log_row position

3dcc66b

Fix R2Solver maxIter constructor

2c4b992

MaxenceGollier force-pushed the RegularizedModels branch from 754feea to 2c4b992 Compare June 15, 2024 10:27

R2 now prints last iteration pass with final stationnarity measure

cff9f50

dpo reviewed Jun 19, 2024

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MaxenceGollier and others added 2 commits June 19, 2024 21:59

Update src/R2_alg.jl

b1ffc31

Co-authored-by: Dominique <[email protected]>

Update src/R2_alg.jl

6dfcb53

Co-authored-by: Dominique <[email protected]>

dpo requested changes Jun 22, 2024

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Update src/R2_alg.jl

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Co-authored-by: Dominique <[email protected]>

Solve nlp.meta.x0 bug

f0a273c

MaxenceGollier added 2 commits July 3, 2024 13:47

Minor changes in R2

95b7106

Update documentation

d088f8c

MaxenceGollier commented Jul 3, 2024

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L2 Penalty Algorithm #143

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dpo merged commit e4c69af into JuliaSmoothOptimizers:master Aug 15, 2024
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MaxenceGollier deleted the RegularizedModels branch August 20, 2024 09:10

MaxenceGollier mentioned this pull request Nov 13, 2024

Augmented Lagrangian method #158

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JSO Compliance: R2 #139

JSO Compliance: R2 #139

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	ψ(s; xₖ) is either h(xₖ + s) or an approximation of h(xₖ + s), ‖⋅‖ is a user-defined norm and σₖ > 0 is the regularization parameter.
	ψ(s; xₖ) is either h(xₖ + s) or an approximation of h(xₖ + s), ‖⋅‖ is a user-defined norm and σₖ > 0 is the regularization parameter.

JSO Compliance: R2 #139

JSO Compliance: R2 #139

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