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Some popular functions used to test global optimization algorithms havemultiple local optima, all with the same value. That is all local optima arealso global optima. This paper suggests that such functions are easilyfortified by adding a localized bump at the location of one of the optima,making the functions more difficult to optimize due to the multiple competinglocal optima. This process is illustrated here for the Branin-Hoo function,which has three global optima. We use the popular Python SciPy differentialevolution (DE) optimizer for the illustration. DE also allows the use of thegradient-based BFGS local optimizer for final convergence. By making a largenumber of replicate runs we establish the probability of reaching a globaloptimum with the original and fortified Branin-Hoo. With the original functionwe find 100% probability of success with a moderate number of functionevaluations. With the fortified version, the probability of getting trapped ina non-global optimum could be made small only with a much larger number offunction evaluations. However, since the probability of ending up at the globaloptimum is usually 1/3 or more, it may be beneficial to perform multipleinexpensive optimizations rather than one expensive optimization. Then theprobability of one of them hitting the global optimum can be made high. Wefound that for the most challenging global optimum, multiple runs reducedsubstantially the extra cost for the fortified function compared to theoriginal Branin-Hoo.

Fortified Test Functions for Global Optimization and the Power of Multiple Runs