###### Abstract
According to the \"No Free Lunch\" (NFL) theorem, no algorithm can consistently dominate all optimization problems. Therefore, in scenarios containing both continuous and discrete data, selecting appropriate optimization algorithms is particularly important for mining numerical association rules. The Nutcracker Optimization Algorithm (NOA) has been proven to perform well in complex high-dimensional optimization problems, effectively handling multi-objective problems and having a fast convergence speed [18]. In view of this, this article proposes a multi-objective Nutcracker Optimization Algorithm (C-MONOA) based on a cubic chaotic map, aiming to solve the problem of mining association rules in mixed datasets. C-MONOA combines the chaotic map and Michigan rule encoding method, enabling it to handle mixed data simultaneously. A chaotic map, with its nonlinear and periodic properties, can generate more complex and random sequences and is widely used in image encryption algorithms and swarm intelligence algorithms [19,20]. The cubic chaotic map introduces nonlinear dynamic systems, enhancing population diversity and effectively preventing algorithms from falling into local optima [21]. In addition, the global exploration ability of NOA significantly improves its convergence on complex datasets. Combined with Michigan rule encoding, C-MONOA can directly mine numerical association rules on mixed datasets without the need for discretization. The algorithm takes support, confidence, the Kulc metric, and comprehensibility as the objective functions. Support measures the frequency of itemsets occurring in the dataset and is used to mine frequent association rules; confidence measures the accuracy of association rules, helping to mine accurate strong association rules; Kulc measures the balance between the antecedent and consequent of evaluation rules, which helps to discover more balanced association rules; comprehensibility reflects whether the rules are easy to understand and interpret and is employed for mining simple and highly interpretable association rules. The following are this article's primary contributions:  
1. For the first time, the NOA is introduced into the field of numerical association rule mining on mixed datasets, filling the gap in existing algorithms for handling complex data structures. Meanwhile, optimizing NOA through a cubic chaotic map significantly improves the performance and convergence speed of the algorithm in multidimensional data environments. 2. Innovatively applying the four important indicators of support, confidence, Kulc, and comprehensibility to the evaluation framework of numerical association rule mining provides a more comprehensive evaluation system. By introducing the Pareto dominance theory, we can effectively balance these four objective functions and identify and optimize Pareto optimal solutions. This method ensures that the potential game between various indicators can be reasonably resolved in multi-objective optimization, thereby improving the overall performance and practicality of numerical association rule mining. 3. Use real datasets to validate and evaluate the proposed algorithm, ensuring that it can mine strong association rules with high frequency, high balance, and high comprehensibility. C-MONOA has been compared with other common numerical association rule mining optimization algorithms such as multi-objective particle swarm optimization (MOPSO), multi-objective mantis search algorithm integrated with opposition-based learning and simulated annealing (MOBMSASA), multi-objective whale optimization algorithm (MOWOA), multi-objective firefly algorithm (MOFA), and multi-objective nutctracker optimization algorithm (MONOA).