A matrix-free augmented lagrangian algorithm with application to large-scale structural design optimization

In many large engineering design problems, it is not computationally feasible or realistic to store Jacobians or Hessians explicitly. Matrix-free implementations of standard optimization methods—implementations that do not explicitly form Jacobians and Hessians, and possibly use quasi-Newton approximations—circumvent those restrictions, but such implementations are virtually non-existent. We develop a matrix-free augmented-Lagrangian algorithm for nonconvex problems with both equality and inequality constraints. Our implementation is developed in the Python language, is available as an open-source package, and allows for approximating Hessian and Jacobian information.We show that our approach solves problems from the CUTEr and COPS test sets in a comparable number of iterations to state-of-the-art solvers. We report numerical results on a structural design problem that is typical in aircraft wing design optimization. The matrix-free approach makes solving problems with thousands of design variables and constraints tractable, even when function and gradient evaluations are costly.