Application of the simulated annealing and downhill simplex methods to the design of electrostatic monopole-and-quadrupole circular plate systems is investigated. The design objective is to minimize chromatic, relativistic, and third- and fifth-order geometric aberrations subjected to a set of predefined source parameters, source and image locations, and maximum allowed electrode potentials. Instead of searching for some optimal potential distributions, which may be extremely difficult to reconstruct in practice, a geometrically simple system consisting of circular plates is arbitrarily chosen at the start. These circular plate electrodes are each partitioned into four separate sectors so that a mixture of axially symmetric (monopole) and quadrupole fields can be generated by each individual plate. An algorithm for rapid repeated computations of the field and potential distributions is outlined. Methods of simulated annealing and downhill simplex are then invoked to find a locally optimal monopole and quadrupole potentials combination to achieve the smallest image spot size of an axial point source. Several promising results have been observed. The combination of monopoles and quadrupoles can achieve spot sizes more than two times smaller in radius than those obtained with just monopoles or quadrupoles alone. Finally, the patterns and properties of the optimal monopole and quadrupole potentials are studied in detail to gain some insight into the ideal general system behaviors and interactions.