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.

多目标优化的研究中，提出了一种新颖的 Tchebycheff 集标量化方法，用于找到少量代表性解来覆盖大量目标，并进一步发展出具有良好理论保证的平滑 Tchebycheff 集标量化方法，我们通过在不同问题中进行实验研究来证明我们方法的有效性。

少对多：用切比雪夫集标量化的多目标优化

Few for Many: Tchebycheff Set Scalarization for Many-Objective  Optimization

Multi-objective optimization problems can be found in many real-world
applications, where the objectives often conflict each other and cannot be
optimized by a single solution. In the past few decades, numerous methods have
been proposed to find Pareto solutions that represent different optimal
trade-offs among the objectives for a given problem. However, these existing
methods could have high computational complexity or may not have good
theoretical properties for solving a general differentiable multi-objective
optimization problem. In this work, by leveraging the smooth optimization
technique, we propose a novel and lightweight smooth Tchebycheff scalarization
approach for gradient-based multi-objective optimization. It has good
theoretical properties for finding all Pareto solutions with valid trade-off
preferences, while enjoying significantly lower computational complexity
compared to other methods. Experimental results on various real-world
application problems fully demonstrate the effectiveness of our proposed
method.

本文通过平滑优化技术，提出了一种新颖且轻量级的光滑 Tchebycheff 标量化方法，用于基于梯度的多目标优化问题，具有良好的理论性质，能够以较低的计算复杂度找到符合有效权衡偏好的所有 Pareto 解。实验结果充分证明了我们提出方法的有效性。

多目标优化中的平滑切比雪夫标量化

Smooth Tchebycheff Scalarization for Multi-Objective Optimization

Most of existing neural methods for multi-objective combinatorial
optimization (MOCO) problems solely rely on decomposition, which often leads to
repetitive solutions for the respective subproblems, thus a limited Pareto set.
Beyond decomposition, we propose a novel neural heuristic with diversity
enhancement (NHDE) to produce more Pareto solutions from two perspectives. On
the one hand, to hinder duplicated solutions for different subproblems, we
propose an indicator-enhanced deep reinforcement learning method to guide the
model, and design a heterogeneous graph attention mechanism to capture the
relations between the instance graph and the Pareto front graph. On the other
hand, to excavate more solutions in the neighborhood of each subproblem, we
present a multiple Pareto optima strategy to sample and preserve desirable
solutions. Experimental results on classic MOCO problems show that our NHDE is
able to generate a Pareto front with higher diversity, thereby achieving
superior overall performance. Moreover, our NHDE is generic and can be applied
to different neural methods for MOCO.

通过提出一种新的具有多样性增强的神经启发式方法，我们能够从多目标组合优化问题的不同视角产生更多的 Pareto 解，并在经典的多目标组合优化问题上取得了更高的多样性 Pareto 前沿和卓越的整体性能。