`missing_second_order_ad` warning fires even when an analytical Hessian is supplied via `OptimizationFunction(f; hess = ...)`

[gragusa]

Description

The missing_second_order_ad verbosity message warns that a Hessian-requiring solver was selected without a SecondOrder AD type, and that a SecondOrder fallback "will be created". The check that emits it, however, only inspects prob.f.adtype and the solver's requirements — it never checks whether the OptimizationFunction already carries a user-supplied analytical Hessian:

https://github.com/SciML/Optimization.jl/blob/main/lib/OptimizationBase/src/cache.jl#L66-L76

if !(
        prob.f.adtype isa DifferentiationInterface.SecondOrder ||
            prob.f.adtype isa AutoZygote
    ) &&
        (
        SciMLBase.requireshessian(opt) || SciMLBase.requiresconshess(opt) ||
            SciMLBase.requireslagh(opt)
    )
    @SciMLMessage(
        lazy"The selected optimization algorithm requires second order derivatives, but `SecondOrder` ADtype was not provided. ...",
        processed_verbose, :missing_second_order_ad
    )

So the warning fires for the fully-supported pattern "analytical grad + analytical hess + NoAD", where it is a false positive on both counts:

• Nothing is missing. With NoAD, instantiate_function simply wraps the user callbacks (lib/OptimizationBase/src/function.jl, the SciMLBase.NoAD methods), and on the DI path AD Hessians are only constructed when f.hess === nothing (lib/OptimizationBase/src/OptimizationDIExt.jl, hess = if h == true && f.hess === nothing ...). The user Hessian always takes precedence.
• Nothing "will be created." For NoAD, adtype_to_soadtype maps soadtype = NoAD — no SecondOrder wrapper is built, contrary to the message text.

MWE

using Optimization, OptimizationOptimJL

rosenbrock(x, p) = (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2
function rosen_grad!(g, x, p)
    g[1] = -2 * (1 - x[1]) - 400 * x[1] * (x[2] - x[1]^2)
    g[2] = 200 * (x[2] - x[1]^2)
    return nothing
end
function rosen_hess!(H, x, p)
    H[1, 1] = 2 - 400 * (x[2] - 3 * x[1]^2)
    H[1, 2] = -400 * x[1]; H[2, 1] = -400 * x[1]; H[2, 2] = 200.0
    return nothing
end

f = OptimizationFunction(rosenbrock; grad = rosen_grad!, hess = rosen_hess!)
prob = OptimizationProblem(f, zeros(2))
sol = solve(prob, Newton())

emits

┌ Warning: Verbosity toggle: missing_second_order_ad
│  The selected optimization algorithm requires second order derivatives, but `SecondOrder` ADtype was not provided. So a `SecondOrder` with SciMLBase.NoAD() for both inner and outer will be created, this can be suboptimal and not work in some cases so an explicit `SecondOrder` ADtype is recommended.

while converging to (1.0, 1.0) with retcode: Success. Instrumenting rosen_hess! with a counter shows it is called 18 times during this solve — the analytical Hessian is what Newton consumes, exactly as intended.

Why it matters beyond noise

The message is emitted at cache creation, i.e. once per solve call. Workflows that re-solve a cached OptimizationProblem in a hot loop get one warning per solve. In my use case, where I solve a convex dual problem with analytical derivatives once per MCMC likelihood evaluation, this produces 25,011 copies of the warning (11 MB of logs) in a single chain.

Proposed fix

Gate each requirement on the corresponding user-supplied field being absent, e.g.

needs_hess      = SciMLBase.requireshessian(opt) && prob.f.hess === nothing &&
                  prob.f.fgh === nothing && prob.f.hv === nothing
needs_cons_hess = SciMLBase.requiresconshess(opt) && prob.f.cons_h === nothing
needs_lag_hess  = SciMLBase.requireslagh(opt) && prob.f.lag_h === nothing

if !(prob.f.adtype isa DifferentiationInterface.SecondOrder ||
     prob.f.adtype isa AutoZygote) &&
   (needs_hess || needs_cons_hess || needs_lag_hess)
    @SciMLMessage(...)
end

(and analogously for the AutoZygote branch). Happy to open a PR along these lines if the approach sounds right.

Workaround for anyone hitting this

solve(prob, Newton(); verbose = OptimizationVerbosity(missing_second_order_ad = SciMLLogging.Silent())) silences exactly this toggle while keeping other diagnostics.

Environment

• Optimization v5.6.1, OptimizationBase v5.1.3 (same check present on current main)
• OptimizationOptimJL v0.4.14
• Julia 1.12.6, macOS (aarch64)