Add tutorial accessing Linear and Nonlinear Constraints

tutorials/linear-api/Project.toml

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+[deps]
+NLPModels = "a4795742-8479-5a88-8948-cc11e1c8c1a6"
+NLPModelsTest = "7998695d-6960-4d3a-85c4-e1bceb8cd856"
+LinearOperators = "5c8ed15e-5a4c-59e4-a42b-c7e8811fb125"
+
+[compat]
+NLPModels = "0.20"
+NLPModelsTest = "0.9"
+LinearOperators = "2.2"

tutorials/linear-api/index.jmd

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+---
+title: "Accessing Linear and Nonlinear Constraints (Linear API)"
+tags: ["models", "linear", "constraints", "linear_api"]
+author: "arnavk23"
+---
+
+# Accessing Linear and Nonlinear Constraints (Linear API)
+
+This short tutorial illustrates how to inspect and operate on the constraint Jacobian using the "linear API" utilities available in the NLPModels ecosystem. The linear API is useful when a problem contains both linear and nonlinear constraints and you want to test or operate on the Jacobian and related linear operators without materializing dense matrices.
+
+We demonstrate how to:
+
+- evaluate the constraint vector;
+- obtain the Jacobian sparsity and coordinates;
+- build the `LinearOperator` representation of the Jacobian and use `mul!` to compute Jacobian-vector products; and
+- compare direct `jprod!`/`jtprod!` evaluations with the operator-based `mul!`.
+
+## Example using a test problem
+
+We use a problem from `NLPModelsTest` which may contain both linear and nonlinear constraints. The `NLPModelsTest` package exposes a collection of example problems useful for experimenting with the APIs.
+
+```julia
+using NLPModelsTest, NLPModels, LinearOperators, LinearAlgebra
+
+# Load a test problem (choose one that has constraints)
+list = NLPModelsTest.nlp_problems
+nlp = eval(Symbol(list[7]))()
+
+# Point at which to evaluate
+x = nlp.meta.x0
+
+# Evaluate constraints
+c = zeros(nlp.meta.ncon)
+cons!(nlp, x, c)
+println("c = ", c)
+
+# Get Jacobian sparsity and values
+rows = zeros(Int, nlp.meta.nnzj)
+cols = zeros(Int, nlp.meta.nnzj)
+jac_structure!(nlp, rows, cols)
+vals = zeros(Float64, nlp.meta.nnzj)
+jac_coord!(nlp, x, vals)
+
+println("Jacobian nonzero pattern (first 10):")
+println(hcat(rows[1:min(end,10)], cols[1:min(end,10)], vals[1:min(end,10)]))
+
+# Build a LinearOperator for the Jacobian (operator acts on variable-space vectors)
+Jv = zeros(nlp.meta.ncon)
+Jtv = zeros(nlp.meta.nvar)
+J = jac_op!(nlp, x, Jv, Jtv) # returns a LinearOperator representing the Jacobian
+
+# Compare operator-based J*v with jprod!
+v = randn(nlp.meta.nvar)
+Jv_op = similar(Jv)
+mul!(Jv_op, J, v)               # operator-based product
+Jv_direct = zeros(nlp.meta.ncon)
+jprod!(nlp, x, v, Jv_direct)    # direct Jacobian-vector product API
+
+println("||J*v (op) - J*v (jprod!)|| = ", norm(Jv_op - Jv_direct))
+
+# And similarly for J'*w
+w = randn(nlp.meta.ncon)
+Jt_w_op = zeros(nlp.meta.nvar)
+mul!(Jt_w_op, adjoint(J), w)
+Jt_w_direct = zeros(nlp.meta.nvar)
+jtprod!(nlp, x, w, Jt_w_direct)
+println("||J' * w (op) - J' * w (jtprod!)|| = ", norm(Jt_w_op - Jt_w_direct))
+```
+
+## Notes
+
+- The `jac_op!`/`hess_op!` helpers produce `LinearOperator` objects (see `LinearOperators.jl`), which allow efficient `mul!` operations without assembling dense matrices.
+- The testing utilities in this repository expose a `linear_api` option (for example in `test_allocs_nlpmodels`) that enables checking the linear-specific functions; see the `Test allocations of NLPModels` tutorial for details.