Add affine set and box section projections from JAXopt#1701
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Adds projection_affine_set and projection_box_section to optax.projections, two of the remaining items in issue 1280. Unlike their JAXopt counterparts, neither requires a QP solver: the affine set projection uses the closed-form KKT solution, and the box section projection finds the scalar dual variable by bisection wrapped in jax.lax.custom_root so it is differentiable by implicit differentiation. projection_polyhedron is left out since it depends on an OSQP-style solver, pending issue 977. Includes tests for feasibility, optimality, idempotence, consistency with projection_hyperplane and projection_simplex, and gradient correctness.
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Fixes #1280.
This adds
projection_affine_setandprojection_box_section, two of thethree remaining projections from the checklist.
It turns out neither needs to wait on the solver decision in #977. JAXopt's
projection_affine_setdelegates toEqualityConstrainedQP, but theequality-constrained projection has the closed-form KKT solution
p = x + aᵀ (a aᵀ)⁻¹ (b − a x), which is what's implemented here.projection_box_sectiononly needs the root of a monotone scalar function,so the bisection is done internally (the bracket
[min((lower − x)/w), max((upper − x)/w)]contains the root whenever theproblem is feasible) and is wrapped in
jax.lax.custom_root, making theprojection differentiable by implicit differentiation of the root. Both work
under
jit,vmap, andgrad. Following optax conventions, JAXopt'shyperparamstuples are flattened into named arguments, matchingprojection_hyperplaneandprojection_box.projection_polyhedronis the only item that genuinely requires anOSQP-class solver, so it is left out pending #977.
Tests check feasibility, optimality (residual orthogonal to the constraint
nullspace), idempotence, consistency with
projection_hyperplane(single-rowa) andprojection_simplex(unit weights, inactive upper bound), behaviorunder
jit, and gradients against finite differences. I also verified bothfunctions against a scipy SLSQP oracle locally (agreement ~1e-15 in float64).