Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., $>100$) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method.

翻译：多目标优化广泛存在于现实应用中，其中某些相互冲突的目标无法通过单一解进行优化。现有优化方法通常专注于寻找一组帕累托解，这些解在目标间具有不同的最优权衡。然而，要良好近似整个帕累托最优集所需的解数量可能随目标数呈指数级增长，这使得这些方法难以处理大量优化目标。在本工作中，我们不再寻找密集的帕累托解集，而是提出一种新颖的切比雪夫集合标量化方法，通过协作互补的方式寻找少量代表性解（例如5个）来覆盖大量目标（例如$>100$。通过这种方式，每个目标都能在小型解集中至少被一个解充分处理。此外，我们进一步提出一种平滑的切比雪夫集合标量化方法，该方法在具有良好理论保证的同时能实现高效优化。在多个具有大量优化目标问题上的实验研究验证了所提方法的有效性。