多目标优化问题的范数型和P-S型标量化方法

多目标优化是最优化领域中一个十分重要的方向，其理论与方法研究受到广泛关注，具有广泛的应用前景。标量化方法是指通过恰当的方法将多目标优化问题转化为单目标优化问题，进而通过对单目标优化问题的研究达到研究原多目标优化问题的一类基本方法。范数型标量化和P-S型标量化是多目标优化问题标量化的两类基本方法。本项目重点研究多目标优化问题范数型标量化和P-S型标量化，包括提出新的范数型标量化和P-S型标量化模型，进而建立多目标优化问题各类精确与近似解的标量化刻画结果；同时，针对机器学习中的双目标正则化模型，利用范数型标量化和P-S型标量化方法建立这类特殊多目标优化问题解的标量化结果，为正则化模型的算法设计提供新的思路。本项目的实施不仅能够促进多目标优化理论与方法的进一步发展，也将为应用多目标优化方法解决应用问题提供方法支撑。

Multiobjective optimization is a very important direction in the field of optimization. Study on its theory and method has attracted widespread attention and has a wide range of application prospects. Scalarization method refers to a basic method of transforming a multiobjective optimization problem into a single objective optimization problem by using an appropriate method, and then through the study of a single objective optimization problem to study the original multiobjective optimization problem. Norm-type scalarization and P-S-type scalarization are two basic methods of scalarization for multiobjective optimization problems. This project focuses on norm-type scalarization and P-S-type scalarization of multiobjective optimization problems, including proposing new norm-type scalarization and P-S-type scalarization models, and then establishes some scalarization results of various exact and approximate solutions for multiobjective optimization problems; At the same time, aiming at the biobjective regularization model in machine learning, the norm-type scalarization and P-S-type scalarization methods are used to establish the scalarization results of the solutions for such special multiobjective optimization problems, which can provide new ideas of the algorithm design for the regularization model. The implementation of this project can not only promote the further development of multiobjective optimization theory and methods, but also provide method support for applying multiobjective optimization methods to solve some application problems.