NUMERICAL SENSITIVITY COMPUTATION FOR DISCONTINUOUS GRADIENT-ONLY OPTIMIZATION PROBLEMS USING THE COMPLEX-STEP METHOD

This study considers the numerical sensitivity calculation for discontinuous gradient- only optimization problems using the complex-step method. The complex-step method was initially introduced to differentiate analytical functions in the late 1960s, and is based on a Taylor series expansion using a pure imaginary step. The complex-step method is not subject to subtraction errors as with finite difference approaches when computing first order sensitiv- ities and therefore allows for much smaller step sizes that ultimately yields accurate sensitivi- ties. This study investigates the applicability of the complex-step method to numerically com- pute first order sensitivity information for discontinuous optimization problems. An attractive feature of the complex-step approach is that no real difference step is taken as with conven- tional finite difference approaches, since conventional finite differences are problematic when real steps are taken over a discontinuity. We highlight the benefits and disadvantages of the complex-step method in the context of discontinuous gradient-only optimization problems that result from numerically approximated (partial) differential equations. Gradient-only optimization is a recently proposed alternative to mathematical pro- gramming for solving discontinuous optimization problems. Gradient-only optimization was initially proposed to solve shape optimization problems that utilise remeshing (i.e. the mesh topology is allowed to change) between design updates. Here, changes in mesh topology result in abrupt changes in the discretization error of the computed response. These abrupt changes in turn manifests as discontinuities in the numerically computed objective and constraint func- tions of an optimization problem. These discontinuities are in particular problematic when they manifest as local minima. Note that these potential issues are not limited to problems in shape optimization but may be present whenever (partial) differential equations are ap- proximated numerically with non-constant discretization methods e.g. remeshing of spatial domains or automatic time stepping in temporal domains.