An important aspect of optimization algorithms, for instance evolutionary algorithms, are termination criteria that measure the proximity of the found solution to the optimal solution set. A frequently used approach is the numerical verification of necessary optimality conditions such as the Karush–Kuhn–Tucker (KKT) conditions. In this paper, we present a proximity measure which characterizes the violation of the KKT conditions. It can be computed easily and is continuous in every efficient solution. Hence, it can be used as an indicator for the proximity of a certain point to the set of efficient (Edgeworth-Pareto-minimal) solutions and is well suited for algorithmic use due to its continuity properties. This is especially useful within evolutionary algorithms for candidate selection and termination, which we also illustrate numerically for some test problems.

优化算法（例如进化算法）的重要方面是终止标准，该标准衡量找到的解决方案与最优解决方案集的接近度。一种常用的方法是对必要的最优条件（例如Karush–Kuhn–Tucker（KKT）条件）进行数值验证。在本文中，我们提出了一种接近度量，该度量描述了违反KKT条件的特征。它可以轻松计算，并且在每个有效解决方案中都是连续的。因此，它可以用作某个点与有效（Edgeworth-Pareto-minimal）解集的接近程度的指标，并且由于其连续性而非常适合算法使用。这在用于候选人选择和终止的进化算法中特别有用，