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Why are there so many system shapes in lens design?

The presence of many local minima in the merit function landscape is perhaps the most difficult challenge in lens design. We present a simplified mathematical model that illustrates why the number of local minima increases rapidly with each additional lens added to the imaging system. Comparisons with results obtained with lens design software are made for the design landscape of triplets with variable curvatures, a problem that is nontrivial, but still simple enough to be analyzed in detail. The mathematical model predicts how many types of local minima can exist in the landscape of the global optimization problem and what are, roughly, their curvatures. This model is mathematically quite general and might perhaps be useful as an analogy for understanding other global optimization problems as well, there where the number of local minima increases rapidly when more components of the same kind are added in the model of the problem.