A hole with a given size is placed in the interior of a plate with an arbitrary external boundary. To avoid stress concentrations the shape of the hole must be smooth (continuous curvature). The objectives of the optimization are the eigenfrequencies of the plate with the hole. The optimization is performed in relation to maximizing the first eigenfrequency or maximizing the gap between the first and second eigenfrequency. An inverse problem is also shown, i.e., find the shape and position of a hole in the plate that result in a specified eigenfrequency. To obtain a smooth boundary of the hole we use an analytical description of the hole. A rather general parameterization with only seven design parameters is applied, including the possibility of going from an ellipse to a square or even to a triangle. Optimal designs are obtained iteratively using mathematical programming, each of the redesigns is based on finite element analysis and sensitivity analysis. Mindlin plate theory is the basis for the FE-analysis and the semi-analytical sensitivity analysis includes only the elements on the boundary of the hole.