Geometric Nonlinear Programming Test Problems

Convex Nonlinear Programming problems are all alike; every nonconvex problem is difficult in its own way. I am not the first numerical analyst to borrow the most quoted line of Anna Karenina to highlight the difficulties of non-linearity or non-convexity1. It could be argued that convex problems are not really “all alike”, but happy families are not either, therefore both the famous first sentence of Tolstoi’s novel and its mathematical corrupted version evoke the way in which diversity of obstacles increases, as far as landscapes become more complex. Practical optimizers face this dilemma when they need to evaluate nonlinear optimization algorithms. “Normal” theory, based on the behavior of bounded subsequences and on local convergence rates, is not enough to predict efficiency and reliability of a method. As a consequence, many “plausibility” arguments are many times invoked to justify the introduction of new ideas. For example, as in a golf