WIP: Objective starting values

[blegat]

Proof of concept solution for #3270 copy-pasted from src/copy_dual.jl of jump-dev/DiffOpt.jl#231
Most of it should be in MOI which is why it's more of a POC PR.
The example of #3270 works after this PR:

julia> using SCS, JuMP

julia> model = Model(SCS.Optimizer);

julia> @variable(model, x);

julia> @objective(model, Min, x^2);

julia> optimize!(model)
------------------------------------------------------------------
	       SCS v3.2.1 - Splitting Conic Solver
	(c) Brendan O'Donoghue, Stanford University, 2012
------------------------------------------------------------------
problem:  variables n: 2, constraints m: 3
cones: 	  q: soc vars: 3, qsize: 1
settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
	  acceleration_lookback: 10, acceleration_interval: 10
lin-sys:  sparse-direct-amd-qdldl
	  nnz(A): 3, nnz(P): 0
------------------------------------------------------------------
 iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
------------------------------------------------------------------
     0| 2.07e+01  1.00e+00  2.83e+01 -1.41e+01  1.00e-01  9.90e-05 
    25| 1.14e-05  2.44e-08  1.61e-05  8.04e-06  1.00e-01  1.18e-04 
------------------------------------------------------------------
status:  solved
timings: total: 1.19e-04s = setup: 7.40e-05s + solve: 4.48e-05s
	 lin-sys: 3.28e-06s, cones: 2.97e-06s, accel: 9.37e-07s
------------------------------------------------------------------
objective = 0.000008
------------------------------------------------------------------

julia> set_start_values(model)

julia> model.moi_backend.optimizer.model.model_cache.varattr
Dict{MathOptInterface.AbstractVariableAttribute, Dict{MathOptInterface.VariableIndex, Any}} with 1 entry:
  VariablePrimalStart() => Dict(VariableIndex(2)=>0.0, VariableIndex(1)=>0.0)

julia> model.moi_backend.optimizer.model.model_cache.conattr
Dict{MathOptInterface.AbstractConstraintAttribute, Dict{MathOptInterface.ConstraintIndex, Any}} with 2 entries:
  ConstraintDualStart()   => Dict(ConstraintIndex{VectorAffineFunction{Float64}, SecondOrderCone}(1)=>[0.707107, -0.707107, -0.0])
  ConstraintPrimalStart() => Dict(ConstraintIndex{VectorAffineFunction{Float64}, SecondOrderCone}(1)=>[0.707107, 0.707107, 0.0])

julia> optimize!(model)
------------------------------------------------------------------
	       SCS v3.2.1 - Splitting Conic Solver
	(c) Brendan O'Donoghue, Stanford University, 2012
------------------------------------------------------------------
problem:  variables n: 2, constraints m: 3
cones: 	  q: soc vars: 3, qsize: 1
settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
	  acceleration_lookback: 10, acceleration_interval: 10
lin-sys:  sparse-direct-amd-qdldl
	  nnz(A): 3, nnz(P): 0
------------------------------------------------------------------
 iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
------------------------------------------------------------------
     0| 2.61e-15  2.22e-16  3.55e-15 -1.78e-15  1.00e-01  5.50e-05 
------------------------------------------------------------------
status:  solved
timings: total: 5.61e-05s = setup: 4.21e-05s + solve: 1.40e-05s
	 lin-sys: 6.56e-07s, cones: 1.33e-06s, accel: 2.90e-08s
------------------------------------------------------------------
objective = -0.000000
------------------------------------------------------------------

[codecov]

Codecov Report

Patch coverage: 13.79% and project coverage change: -0.52 ⚠️

Comparison is base (0919995) 98.10% compared to head (904b391) 97.59%.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #3271      +/-   ##
==========================================
- Coverage   98.10%   97.59%   -0.52%     
==========================================
  Files          34       34              
  Lines        4704     4733      +29     
==========================================
+ Hits         4615     4619       +4     
- Misses         89      114      +25     

Impacted Files	Coverage Δ
src/optimizer_interface.jl	88.42% <13.79%> (-8.46%)	⬇️

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[odow]

I dunno, this feels too niche. Couldn't the Objective.SlackBridge have a final_touch that checked for VariablePrimalStart and set any ConstraintPrimalStart/ConstraintDualStart appropriately?

[blegat]

These attributes could be defined in the bridge itself. Doing it in final_touch is weird. It's confusing to have starting values set without the user asking for it.
If users use this JuMP function they it will all be under the hood

[odow]

Doing it in final_touch is weird

Why? The start value depends on other parts of the model that may not be present at the time of the caller. That's exactly why we added final_touch.

If users use this JuMP function they it will all be under the hood

It seems weird to have a JuMP function for a very specific part of MOI (MOI.Bridges.Objective.SlackBridge).

[blegat]

Why? The start value depends on other parts of the model that may not be present at the time of the caller. That's exactly why we added final_touch.

Because you will have starting values set to the solver even if you didn't set any

It seems weird to have a JuMP function for a very specific part of MOI (MOI.Bridges.Objective.SlackBridge).

We wouldn't have a JuMP function just for this. set_start_values would just take care of setting these attributes that would be defined in MOI (possible even inside the bridges

[odow]

If jump-dev/MathOptInterface.jl#2194 is merged, this PR would become

MOI.set(model, MOI.Bridges.Objective.SlackBridgePrimalDualStart(), true)

[odow]

Closing for now. Once the MOI PR is merged we can look at this again. The issue will stay open.