One-dimensional Optimization on Non-archimedean Fields

One dimensional optimization on non-Archimedean fields is presented. We derive first and second order necessary and sufficient optimality conditions. For first order optimization, these conditions are similar to the corresponding real ones; but this is not the case for higher order optimization. This is due to the total disconnectedness of the given non-Archimedean field in the order topology, which renders the usual concept of differentiability weak. We circumvent this difficulty by using a stronger concept of differentiability based on the derivate approach, which entails a Taylor formula with remainder and hence a similar local behavior as in the real case.