Add ConvexOptimization subpackage (disciplined convex programming backend)

[ChrisRackauckas-Claude]

Note

Draft / experimental, and blocked on SciML/SciMLBase.jl#1440 (the ConvexOptimizationProblem/ConvexOptimizationSolution types are unreleased). Please ignore until reviewed by @ChrisRackauckas. CI will be red until #1440 merges and SciMLBase releases — the [sources] git-rev is a temporary bridge.

New lib/ConvexOptimization subpackage: the disciplined-convex-programming backend for the SciML optimization stack, and the first vertical slice of the Convex.jl / cvxpy replacement effort (SymbolicAnalysis.jl#121).

What it does

solve(::ConvexOptimizationProblem, ConvexMOI(Clarabel.Optimizer)):

1. Certifies the objective and constraints convex with SymbolicAnalysis.jl (analyze), erroring (route-to-NLP) if not — a non-convex problem is rejected, never silently mis-solved.
2. Lowers the affine objective and each ConeConstraint to MathOptInterface via Symbolics.linear_expansion.
3. Solves with Clarabel (any MOI conic solver via the optimizer_constructor).
4. Returns a ConvexOptimizationSolution carrying the primal and the dual multipliers — the optimality certificate that OptimizationSolution lacks (OptimizationMOI never surfaces duals).

Purpose of this PR

Primarily to verify the new SciMLBase interface end-to-end — that a ConvexOptimizationProblem constructs, solve routes to a backend, and a ConvexOptimizationSolution with correct duals comes back. It does.

Design notes

• Routing: ConvexOptimizationProblem <: AbstractOptimizationProblem is a sibling of OptimizationProblem, and OptimizationBase's solve/init dispatch on the concrete OptimizationProblem, so the standard indirection is not inherited. The backend therefore defines SciMLBase.init(::ConvexOptimizationProblem, alg) and SciMLBase.solve!(::ConvexOptimizationCache); solve = solve! ∘ init composes them via CommonSolve.
• Constraint representation (the opaque ConvexOptimizationProblem.constraints field): a Vector{ConeConstraint}, each ConeConstraint(g, set) pairing a callable affine map g(u,p)->Vector with an MOI cone. This is symmetric with the callable objective (one tracing path), gives exact 1:1 constraint→dual alignment, and fixes the dual sign at construction (>=→Nonnegatives, <=→Nonpositives, ==→Zeros) so the returned ConstraintDual is already in the user's variables — verified to match Convex.jl with no sign flip.

Scope (MVP)

Affine objective + affine-in-cone constraints (LP), plus second-order-cone constraints. A norm2-in-objective epigraph atom (add aux t, [t; Ax+b] ∈ SOC) and the full symbolic atom→cone canonicalizer are the next steps.

Verification

Pkg.test() locally (against the #1440 SciMLBase branch): Core 15/15 pass, covering

• LP min x₁+2x₂ s.t. x₁+x₂=1, x≥0 → primal [1,0], obj 1, duals [1] (equality) and [0,1] (nonneg cone) — machine-exact vs the analytic values (which equal Convex.jl's);
• an SOC-constraint solve (min t s.t. ‖(x₁,x₂)‖≤t, x=(3,4) → t=5);
• non-convex objective (x₁x₂) rejected by certification;
• the solution cache exposing the SciMLBase u0/p glue fields.

An optional Convex.jl cross-check is included (skipped when Convex isn't in the env, since the analytic assertions already pin the same numbers).

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