## Computationally relevant generalized derivatives: theory, evaluation and applications.(English)Zbl 1401.49016

Summary: A new method for evaluating generalized derivatives in nonsmooth problems is reviewed. Lexicographic Directional (LD-)derivatives are a recently developed tool in nonsmooth analysis for evaluating generalized derivative elements in a tractable and robust way. Applicable to problems in both steady-state and dynamic settings, LD-derivatives exhibit a number of advantages over current theory and algorithms. As highlighted in this article, the LD-derivative approach now admits a suitable theory for inverse and implicit functions, nonsmooth dynamical systems and optimization problems, among others. Moreover, this technique includes an extension of the standard vector forward mode of Automatic Differentiation (AD) and acts as the natural extension of classical calculus results to the nonsmooth case in many ways. The theory of LD-derivatives is placed in the context of state-of-the-art methods in nonsmooth analysis, with an application in multistream heat exchanger modeling and design used to illustrate the usefulness of the approach.

### MSC:

 49J52 Nonsmooth analysis 49M15 Newton-type methods 65K15 Numerical methods for variational inequalities and related problems 90C31 Sensitivity, stability, parametric optimization 90C56 Derivative-free methods and methods using generalized derivatives 49Q12 Sensitivity analysis for optimization problems on manifolds

### Software:

NDA; PNEW; PATH Solver; dcc; libMC
Full Text:

### References:

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