Research On Generalized Convex Programming With Local Solution Being Global Solution

The optimization problem needs to give the optimal solution or the optimal solution set.The single objective programming problem is relatively perfect in theory and application,but in practical application scenarios,most of them are multi-objective programming problems,such as resource allocation problems,investment problems,etc.The difficulty of multi-objective programming problems is that different objectives may conflict with each other,in order to get better on one index,we need to sacrifice another index.So there is no absolute optimal solution for multi-objective programming,which makes every index reach the optimal solution,there is only an effective solution(also known as negotiated solution,Pareto optimal solution).The difficulty of finding the effective solution for multi-objective programming is higher than that for single objective programming.For a convex single objective programming problem,the local optimal solution must be the global optimal solution.Similarly,for a convex multi-objective programming problem,the local effective solution must be the global effective solution.We call it the local solution and the global solution.For a nonconvex programming problem,the local solution is not necessarily the global solution.It has become a hot topic in the field of optimization to discuss the nonconvex programming problem with local solution as global solution.This paper discusses this problem in two ways.One is to start with the nonconvex programming defined by the generalized convex function,the other is to describe this kind of programming problem from the level set of the function definition and lower semicontinuous of the corresponding point-to-set mappings.We will discuss and explore the generalized convex set in detail,the logic thinking of the generalized convex function,and define the more generalized convex set and generalized convex function.In order to construct a more generalized convex programming,we study the programming of local solution as global solution,analyze the mistakes in some literatures,and give simpler counterexamples.We sort out the concepts of level set and lower semicontinuity of numerical function and the ideas of its generalized definition,and sort out the related theorem that local optimal solution is global optimal solution.In this paper,the concept of generalized convex vector valued function based on the definition of generalized convex set is summarized,and simpler examples are given to prove the existence of generalized convex set and generalized convex vector valued function.The definition of the level set of the reference function and the corresponding lower semicontinuous point-to-set mappings are given.The definition of the level set of the vector-valued function and the corresponding lower semicontinuous point-to-set mappings are given.The multi-objective programming of the local efficient solution as the global efficient solution is described.In this paper,we make up for the deficiency of the definition of level set,lower semicontinuity and arc-convexity of vector valued function.We give the proof process of the theorem that the local effective solution is the global effective solution and give examples.