We present a new algorithm for infinite-dimensional optimization with general constraints, called ALESQP. In a nutshell, ALESQP is an augmented Lagrangian method that penalizes inequality constraints and solves equality-constrained nonlinear optimization subproblems at every iteration. The subproblems are solved using a matrix-free trust-region sequential quadratic programming (SQP) method that takes advantage of iterative, i.e., inexact linear solvers and is suitable for PDE-constrained optimization and other large-scale applications.
We analyze convergence of ALESQP under different assumptions. We show that strong accumulation points are stationary, i.e., in finite dimensions ALESQP converges to a stationary point. In infinite dimensions we establish that weak accumulation points are feasible in many practical situations. Under additional assumptions we show that weak accumulation points are stationary.
In the context of optimal control problems, e.g., in PDE-constrained optimization, ALESQP provides a unified framework to efficiently handle general constraints on both the state variables and the control variables. A key algorithmic feature is a constraint decomposition strategy that allows ALESQP to exploit problem-specific variable scalings and inner products. We present several examples with state and control inequality constraints where ALESQP shows remarkable mesh-independent performance, requiring only a handful of outer (AL) iterations to meet constraint tolerances at the level of machine precision. At the same time, ALESQP uses the inner (SQP) loop economically, requiring only a few dozen SQP iterations in total.