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 existing distance metric learning methods assume perfect side
information that is usually given in pairwise or triplet constraints. Instead,
in many real-world applications, the constraints are derived from side
information, such as users' implicit feedbacks and citations among articles. As
a result, these constraints are usually noisy and contain many mistakes. In
this work, we aim to learn a distance metric from noisy constraints by robust
optimization in a worst-case scenario, to which we refer as robust metric
learning. We formulate the learning task initially as a combinatorial
optimization problem, and show that it can be elegantly transformed to a convex
programming problem. We present an efficient learning algorithm based on smooth
optimization [7]. It has a worst-case convergence rate of
O(1/{\surd}{\varepsilon}) for smooth optimization problems, where {\varepsilon}
is the desired error of the approximate solution. Finally, our empirical study
with UCI data sets demonstrate the effectiveness of the proposed method in
comparison to state-of-the-art methods.

本文提出了一种称为鲁棒测度学习的方法，旨在从嘈杂的侧面信息中学习距离度量，将学习任务最初制定为组合优化问题，并将其优雅地转换为凸规划问题，提出了一种基于平滑优化的高效学习算法，最后在 UCI 数据集上的实验研究证明了所提出方法的有效性。