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Many optical design programs use various forms of the damped least-squares method for local optimization. In this paper, we show that damped least-squares algorithms, with maximized computational speed, can create sensitivity with respect to changes in initial conditions. In such cases, starting points, which are very close to each other, lead to different local minima after optimization. Computations of the fractal capacity dimension show that sets of these starting points, which lead to the same minimum (the basins of attraction for that minimum), have a fractal structure. Introducing more damping makes the optimization process stable.

Instabilities and fractal basins of attraction in optical system optimization