We study the minimization of objective functions containing non-physical jump discontinuities. These discontinuities arise when (partial) differential equations are discretized using non-constant methods and the resulting numerical solutions are used in computing the objective function. Although the functions may become discontinuous, gradient information may be computed at every point. Gradient information is computable everywhere since every point has an associated discretization for which (semi) analytical sensitivities can be calculated. Rather than the construction of global approximations using only function value information to overcome the discontinuities, we propose to use only the gradient information. We elaborate on the modifications of classical gradient based optimization algorithms for use in gradient-only approaches, and we then present gradient-only optimization strategies using both BFGS and a new spherical quadratic approximation for sequential approximate optimization (SAO). We then use the BFGS and SAO algorithms to solve three problems of practical interest, both unconstrained and constrained.
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Wilke, D.N., Kok, S. & Groenwold, A.A. The application of gradient-only optimization methods for problems discretized using non-constant methods. Struct Multidisc Optim 40, 433 (2010). https://doi.org/10.1007/s00158-009-0389-x
- Gradient-only optimization
- Step discontinuity
- Partial differential equation
- Non-constant discretization