Refactor approximations into a pure-JuMP layer

[acostarelli]

For the purpose of more easily testing approximations separate from container API. Also includes speedups by using vectorized JuMP object creation. Potentially useful for Data Centers work.

[github-actions]

Performance Results
Main

This branch

[jd-lara]

What's the rationale for this change?

[acostarelli]

Review 1

Overall, don't take this as just nit-picks. I think the major refactor is not done yet. The jump layer still looks like it did with containers just now we're calling them dense axis arrays. I want to be able to read the jump code and cleanly see the formulation without other noise. The name and time axes are fluff that we handle after.

Claude:

Resolved

• Vectorized every register_in_container! copy loop — target.data .= source.data on the underlying Array, no more per-(name, k, t) loops.
• Typed every *Result struct's parameters with concrete bounds (<: QuadraticApproxResult, <: DenseAxisArray{AffExpr, 2}, etc.).
• Removed sparse = true from src/approximations/. Split the generic McCormick into McCormickLowerConstraint / McCormickUpperConstraint so the lower side cleanly disappears when lower_bounds = false; densely-filled containers (HybSBoundConstraint, reformulated McCormick, sawtooth MIP, epigraph tangent) all switched to dense. Also fixed a latent sawtooth schema bug — adjacency container was declared 3D but populated 4D.
• Renamed _add_quadratic_approx! → add_quadratic_approx! and _add_bilinear_approx! → add_bilinear_approx!. POM call sites need the paired rename.
• Converted !== nothing guards to Nothing dispatch for register_reformulated_mccormick!, register_mccormick_envelope!, _register_tightening!, register_pwmcc!, plus a new _register_sawtooth_tightening!.
• Dropped redundant MinMax(...) wrapper (bin2.jl, hybs.jl) and collect(name_axis) throughout.
• Vectorized JuMP-side construction loops: pwmcc_cuts.jl brk, sawtooth MIP (4 vectorized @constraint families instead of a triple loop), epigraph LP and tangent cuts, manual-SOS2 adjacency, and build_mccormick_envelope.
• Extracted scale_back_g_basis(...) in epigraph.jl and reused it from sawtooth and the epigraph tangent cuts.
• Restored precomputed-form add_bilinear_approx! entrypoints for every bilinear method (Bin2, HybS, NMDT, DNMDT, NoBilinear). Backed by a small _PrebuiltQuadApprox adapter in common.jl.
• pwmcc_cuts.jl:144: confirmed JuMP's macro parser recognizes the sum(...) pattern.

Deferred (next round)

• The architectural ask itself — split JuMP math into a single-(name, t)-element routine and broadcast axes on top: bin2.jl:41, mccormick.jl:71, nmdt_discretization.jl:265, plus the abstract-Result base at nmdt_bilinear.jl:242, register-side constraint organization at epigraph.jl:138, common-check wrapper at hybs.jl:59/:123, and loop-count reduction at nmdt_discretization.jl:388.
• epigraph.jl:249 (numeric axis → string axis): downstream to_output_dataframe behavior, skipped for now.

All 1098 IOM tests pass.

[github-actions]

[JuliaFormatter] _{reported by reviewdog 🐶}

Suggested change

	model, x, y,
	psq.approximation, xsq.approximation, ysq.approximation,
	x_min, x_max, y_min, y_max,
	)
	model, x, y,
	psq.approximation, xsq.approximation, ysq.approximation,
	x_min, x_max, y_min, y_max,
	)

[github-actions]

[JuliaFormatter] _{reported by reviewdog 🐶}

Suggested change

	container, ReformulatedMcCormickConstraint, C,
	name_axis, 1:4, time_axis; meta,
	)
	container, ReformulatedMcCormickConstraint, C,
	name_axis, 1:4, time_axis; meta,
	)

[github-actions]

[JuliaFormatter] _{reported by reviewdog 🐶}

Suggested change

	model, x, y, z_var, x_min, x_max, y_min, y_max,
	)
	model, x, y, z_var, x_min, x_max, y_min, y_max,
	)

[github-actions]

[JuliaFormatter] _{reported by reviewdog 🐶}

Suggested change

	container, McCormickConstraint, C, name_axis, 1:4, time_axis; meta,
	)
	container, McCormickConstraint, C, name_axis, 1:4, time_axis; meta,
	)

[github-actions]

[JuliaFormatter] _{reported by reviewdog 🐶}

Suggested change

	container, McCormickUpperConstraint, C,
	name_axis, 1:depth, 1:2, time_axis; meta = mc_meta * "_lb",
	)
	container, McCormickUpperConstraint, C,
	name_axis, 1:depth, 1:2, time_axis; meta = mc_meta * "_lb",
	)

[github-actions]

[JuliaFormatter] _{reported by reviewdog 🐶}

Suggested change

	container, C, name_axis, time_axis, config.epigraph_depth, meta,
	)
	container, C, name_axis, time_axis, config.epigraph_depth, meta,
	)

[github-actions]

[JuliaFormatter] _{reported by reviewdog 🐶}

Suggested change

	container, C, name_axis, time_axis, config.epigraph_depth, meta,
	)
	container, C, name_axis, time_axis, config.epigraph_depth, meta,
	)

[acostarelli]

Closing because probably out of scope.