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Computational complexity of optimization and crude range testing: a new approach motivated by fuzzy optimization

机译：优化和原始范围测试的计算复杂性：一种基于模糊优化的新方法

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摘要

It is often important to test whether the maximum max_B f of a given function f on a given set B is smaller than a given number C. This "crude range testing" (CRT) problem is one of the most important problems in the practical application of interval analysis. Empirical evidence shows that the larger the difference C ― max_B f, the easier the test. In general, the fewer global maxima, the easier the test; and finally, the further away global maxima are from each other, the easier the test. Using standard complexity theory to explain these empirical observations fails because the compared CRT problems are all NP-hard. In this paper the analysis of fuzzy optimization is used to formalize the relative complexity of different CRT problems. This new CRT-specific relative complexity provides a new and "robust" theoretical explanation for the above empirical observations. The explanation is robust because CRT relative complexity takes numerical inaccuracy into consideration. The new explanation is important because it is a more reliable guide than empirical observations to developers of new solutions to the CRT problem.
机译：测试给定集合B上给定函数f的最大max_B f是否小于给定数C常常很重要。此“粗略范围测试”（CRT）问题是实际应用中最重要的问题之一间隔分析。经验证据表明，差异C ― max_B f越大，测试越容易。通常，全局最大值越少，测试越容易；最后，全局最大值之间的距离越远，测试就越容易。使用标准复杂性理论来解释这些经验性观察失败，因为比较的CRT问题都是NP-hard。在本文中，通过对模糊优化的分析来确定不同CRT问题的相对复杂度。这种针对CRT的新的相对复杂性为上述经验观察提供了新的“稳健”的理论解释。这种解释是有力的，因为CRT的相对复杂性考虑了数值误差。新的解释很重要，因为它比对CRT问题新解决方案的开发人员的经验观察更可靠。

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