This paper deals with the resolutions of fuzzy relation equations with
addition-min composition. When the fuzzy relation equations have a solution, we
first propose an algorithm to find all minimal solutions of the fuzzy relation
equations and also supply an algorithm to find all maximal solutions of the
fuzzy relation equations, which will be illustrated, respectively, by numeral
examples. Then we prove that every solution of the fuzzy relation equations is
between a minimal solution and a maximal one, so that we describe the solution
set of the fuzzy relation equations completely.

本文研究了带有加法 - 最小合成的模糊关系方程的解决方案。当模糊关系方程有解时，我们首先提出了一个算法来找到所有模糊关系方程的最小解，并提供了一个算法来找到所有最大解，分别通过数字例子进行说明。然后我们证明了模糊关系方程的每个解都在最小解和最大解之间，使得我们完全描述了模糊关系方程的解集。

具有加法 - 最小合成的模糊关系方程的解集

The solution set of fuzzy relation equations with addition-min composition

Black-box functions are broadly used to model complex problems that provide
no explicit information but the input and output. Despite existing studies of
black-box function optimization, the solution set satisfying an inequality with
a black-box function plays a more significant role than only one optimum in
many practical situations. Covering as much as possible of the solution set
through limited evaluations to the black-box objective function is defined as
the Black-Box Coverage (BBC) problem in this paper. We formalized this problem
in a sample-based search paradigm and constructed a coverage criterion with
Confusion Matrix Analysis. Further, we propose LAMBDA (Latent-Action
Monte-Carlo Beam Search with Density Adaption) to solve BBC problems. LAMBDA
can focus around the solution set quickly by recursively partitioning the
search space into accepted and rejected sub-spaces. Compared with La-MCTS,
LAMBDA introduces density information to overcome the sampling bias of
optimization and obtain more exploration. Benchmarking shows, LAMBDA achieved
state-of-the-art performance among all baselines and was at most 33x faster to
get 95% coverage than Random Search. Experiments also demonstrate that LAMBDA
has a promising future in the verification of autonomous systems in virtual
tests.

本文提出了一种基于样本搜索范式的黑箱覆盖问题及其解决方法 LAMBDA，该方法通过引入密度信息来克服优化采样偏差，其性能在基准测试中表现优异，可用于验证虚拟测试中的自主系统。