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K. Sturm:
"First-order differentiability properties of a class of equality constrained optimal value functions with applications to shape optimization";
Journal of Nonsmooth Analysis and Optimization, 1 (2020).

In this paper we study the right differentiability of a parametric in mum function over a
parametric set defined by equality constraints. We present a new theorem with sufficient conditions for the right differentiability with respect to the parameter. Target applications are nonconvex
objective functions with equality constraints arising in optimal control and shape optimisation.
The theorem makes use of the averaged adjoint approach in conjunction with the variational
approach of Kunisch, Ito and Peichl. We provide two examples of our abstract result: (a) a shape
optimisation problem involving a semilinear partial differential equation which exhibits infinitely
many solutions, (b) finite dimensional quadratic function subject to a nonlinear equation.

In this paper we study the right differentiability of a parametric in mum function over a
parametric set defined by equality constraints. We present a new theorem with sufficient conditions for the right differentiability with respect to the parameter. Target applications are nonconvex
objective functions with equality constraints arising in optimal control and shape optimisation.
The theorem makes use of the averaged adjoint approach in conjunction with the variational
approach of Kunisch, Ito and Peichl. We provide two examples of our abstract result: (a) a shape
optimisation problem involving a semilinear partial differential equation which exhibits infinitely
many solutions, (b) finite dimensional quadratic function subject to a nonlinear equation.

Nonsmooth optimization

http://dx.doi.org/10.46298/jnsao-2020-6034

https://publik.tuwien.ac.at/files/publik_293041.pdf