An algorithm for finding the extreme values of a function of many variables and its application in the automatic design of astronomical optical systems

It is shown that if the equipotentials are n-dimensional ellipsoids, then the streamlines will converge to the centre of the ellipsoids representing minimum potential, and in doing so, the streamlines will get closer and closer to the longest axist of the ellipsoids.

Based on the above consideration, an algorithm is presented for finding the minimum point of the function φ = Σf_i². At each iteration, we take the linear expansions of f_i and proceed along the streamlines corresponding to the resulting, approximate φ, of which the equipotentials are ellipsoids. When we are close to the longest axis of the ellipsoids, we take a suitable leap along the line joining the current point to the centre of the ellipsoid, to begin the next iteration. This method has been found effective in the automatic design of astronomical optical systems. Three examples are given.