Minimax optimization of shaped reflector antennas using sequential convex programming

Shaped reflector antennas are common on communications satellites because of their low-cost beam-shaping ability. Optimization of the reflector surface is often performed using an iterative process that adjusts coefficients of the surface expansion based on the minimax norm of the far-field directivity. Traditionally this involves sampling the directivity pattern on a fixed uniform grid within a specification polygon. It is noted that this is a problem of nonconvex optimization, and as such a global solution cannot be assured using gradient methods.; A new approach to a solution of the optimization problem is presented that exploits the spatial filtering aspect of the antenna pattern. In this approach the specification polygon is replaced by a sequence of polygons beginning with a circumscribing ellipse and ending with the specification polygon. The problem is solved essentially by a sequence of optimizations, the result of which is a solution that is superior in the minimax sense. A theoretical justification is given which shows that the method, referred to a sequential convex programming (SCP), avoids most of the local optima that can trap the original problem into an inferior solution. Integral to the method is a novel adaptive gridding technique that is used to formulate the objective function and is particularly well suited to minimax optimization.; It is noted that the SCP method exhibits linear convergence throughout most of the process. However, this slow rate of convergence should be acceptable in commercial design applications considering the significant antenna performance advantage that can be achieved.