Equality-Constrained Engineering Design Optimization Using a Novel Inexact Quasi-Newton Method

Abstract

For engineering design optimization, the full-space formulation offers the potential for greater efficiency than the more commonly used reduced-space formulation. This potential is greater when the numerical model involves discretized partial differential equations or coupled disciplines. However, the full-space formulation results in a larger optimization problem with at least a factor of two increase in the number of optimization variables and equality constraints. Using Newton-type methods to solve such problems involves solving a large-scale and, often, ill-conditioned Karush–Kuhn–Tucker linear system at each optimization iteration. This can be time-consuming to solve even with a Krylov solver. If the number of iterations is reduced, the full-space formulation could be applied to a broader class of problems. This paper presents an inexact quasi-Newton algorithm with an adaptive extension for solving large-scale equality-constrained optimization problems. The new algorithm inexactly solves the Karush–Kuhn–Tucker system using new inexactness criteria that are derived to ensure a descent direction. The adaptive extension chooses the stopping condition of the Krylov solver by also taking its convergence rate into account. The paper presents results of numerical experiments applying this algorithm to three types of problems: six constrained optimization problems from the widely used CUTEst test suite, a bar thickness optimization problem, and a two-dimensional topology optimization problem. For all problems, the new algorithm consistently shows a roughly 50% reduction in the total number of Krylov solver iterations and a minimum of roughly 15% reduction in the optimization time. Moreover, the proposed approach for selecting the Krylov solver tolerance shows an improvement in all cases, whereas the existing forcing-parameter approach shows an increase in the number of Krylov iterations in some cases. These results indicate that this new method for selecting solver tolerances is effective and robust, and a good choice in algorithms that use a Krylov solver for solving the Karush–Kuhn–Tucker linear system.