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Minimax Theory and its Applications 06 (2021), No. 1, 127--144
Copyright Heldermann Verlag 2021

Optimality Conditions in Discrete-Continuous Nonlinear Optimization

Gabriele Eichfelder
Institut für Mathematik, TU Ilmenau, 98693 Ilmenau, Germany

Johannes Jahn
Department Mathematik, Universität Erlangen-Nürnberg, 91058 Erlangen, Germany

We present necessary and sufficient optimality conditions for discrete-continuous nonlinear optimization problems including mixed-integer nonlinear problems. This theory does not utilize an extension of the Lagrange theory of continuous optimization but it works with certain max functionals for a separation of two sets where one of them is nonconvex. These functionals have the advantage that they can be used for nonconvex optimization problems. This theory avoids getting several Lagrange multipliers per constraint.

Keywords: Nonlinear optimization, optimality conditions, discrete-continuous variables, mixed-integer nonlinear problems.

MSC: 90C30, 90C46, 90C11.

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