The linear scalarizations and Lagrange multipliers for vector optimization

Abstract

Vector optimization is an important part of mathematical programming. Its theory and methods have a promising interdisciplinary research field with many significantapplications. In this survey, we mainly introduce the progress on the generalized convexity of vector-valued maps, alternative theorems, linear scalarization methods and Lagrange multiplier rules. We first introduce a class of generalized convexity for the vector-valued and set-valued maps, which is based on image space analysis, and summarize the relationship among them. Secondly, we introduce the development of the alternative theorems in linear systems to nonlinear systems. For nonlinear systems, we focus on the research of the alternative theorem under convexity or generalized convexity assumptions. Its applications in the linear scalarization and the Lagrange multiplier rules on vector optimization problems are summarized.