Reference

model = FirstOrderModel(f, ∇f!; name = "first-order model")

A simple subtype of AbstractNLPModel to represent a smooth objective.

Arguments

• f :: F <: Function: a function such that f(x) returns the objective value at x;
• ∇f! :: G <: Function: a function such that ∇f!(g, x) stores the gradient of the objective at x in g.

model = FirstOrderNLSModel(r!, jv!, jtv!; name = "first-order NLS model")

A simple subtype of AbstractNLSModel to represent a nonlinear least-squares problem with a smooth residual.

Arguments

• r! :: R <: Function: a function such that r!(y, x) stores the residual at x in y;
• jv! :: J <: Function: a function such that jv!(u, x, v) stores the product between the residual Jacobian at x and the vector v in u;
• jtv! :: Jt <: Function: a function such that jtv!(u, x, v) stores the product between the transpose of the residual Jacobian at x and the vector v in u.

model, nls_model, sol = bpdn_model(args...)
model, nls_model, sol = bpdn_model(compound = 1, args...)

Return an instance of an NLPModel and an instance of an NLSModel representing the same basis-pursuit denoise problem, i.e., the under-determined linear least-squares objective

½ ‖Ax - b‖₂²,

where A has orthonormal rows and b = A * x̄ + ϵ, x̄ is sparse and ϵ is a noise vector following a normal distribution with mean zero and standard deviation σ.

Arguments

• m :: Int: the number of rows of A
• n :: Int: the number of columns of A (with n ≥ m)
• k :: Int: the number of nonzero elements in x̄
• noise :: Float64: noise standard deviation σ (default: 0.01).

The second form calls the first form with arguments

m = 200 * compound
n = 512 * compound
k =  10 * compound

Return Value

An instance of a FirstOrderModel and of a FirstOrderNLSModel that represent the same basis-pursuit denoise problem, and the exact solution x̄.

fh_model(; kwargs...)

Return an instance of an NLPModel and an instance of an NLSModel representing the same Fitzhugh-Nagumo problem, i.e., the over-determined nonlinear least-squares objective

½ ‖F(x)‖₂²,

where F: ℝ⁵ → ℝ²⁰² represents the fitting error between a simulation of the Fitzhugh-Nagumo model with parameters x and a simulation of the Van der Pol oscillator with fixed, but unknown, parameters.

Keyword Arguments

All keyword arguments are passed directly to the ADNLPModel (or ADNLSModel) constructure, e.g., to set the automatic differentiation backend.

Return Value

An instance of an ADNLPModel that represents the Fitzhugh-Nagumo problem, an instance of an ADNLSModel that represents the same problem, and the exact solution.