PDENLPModels.jl: An NLPModel API for Optimization Problems with PDE-Constraints

Abstract

Shape optimization, optimal control, and parameter estimation of systems governed by partial differential equations (PDE) give rise to a class of problems known as PDE-constrained optimization (Hinze et al., 2008). PDENLPModels.jl is a Julia (Bezanson et al., 2017) package for modeling and discretizing optimization problems with mixed algebraic and PDE in the constraints. The general form of the problems over some domain Ω ⊂ ℝ 𝑑 is minimizewhere 𝑦 ∶ Ω → 𝒴 is the state, 𝑢 ∶ Ω → 𝒰 is the control, and 𝜃 ∈ ℝ 𝑘 are algebraic variables. 𝐽 ∶ 𝒴 × 𝒰 × ℝ 𝑘 → ℝ and 𝑒 ∶ 𝒴 × 𝒰 × ℝ 𝑘 → 𝒞 are smooth mappings. (𝒴, ‖ ⋅ ‖ 𝒴 ), (𝒰, ‖ ⋅ ‖ 𝒰 ), and (𝒞, ‖ ⋅ ‖ 𝒞 ) are real Banach spaces, 𝑙 𝜃 , 𝑢 𝜃 ∈ ℝ 𝑘 are bounds on 𝜃, and 𝑙 𝑦𝑢 , 𝑢 𝑦𝑢 ∶ Ω → 𝒴 × 𝒰 are functional bounds on the controls and states.After discretization of the domain Ω, the integral, and the derivatives, the resulting problem is a nonlinear optimization problem of the form minimize 𝑥∈ℝ 𝑁 𝑦 +𝑁 𝑢 +𝑁 𝜃 𝑓(𝑥) subject to 𝑐(𝑥) = 0, 𝑙 ≤ 𝑥 ≤ 𝑢,