On continuous selections of polynomial functions

作     者：Guo, Feng Jiao, Liguo Sang Kim, Do 

作者机构：Dalian Univ Technol Sch Math Sci Dalian Peoples R China Northeast Normal Univ Acad Adv Interdisciplinary Studies Changchun Jilin Peoples R China Pukyong Natl Univ Dept Appl Math Busan South Korea 

出 版 物：《OPTIMIZATION》 (最优化)

年 卷 期：2024年第73卷第2期

页      面：295-328页

核心收录：

学科分类：12[管理学] 1201[管理学-管理科学与工程(可授管理学、工学学位)] 07[理学] 070105[理学-运筹学与控制论] 0701[理学-数学] 

基　　金：Chinese National Natural Science Foundation Fundamental Research Funds for the Central Universities Natural Science Foundation of Jilin Province [YDZJ202201ZYTS302] National Research Foundation of Korea Grant - Korean Government [NRF-2019R1A2C1008672] 

主　　题：Continuous selections polynomial functions critical points generic properties 

摘      要：A continuous selection of polynomial functions is a continuous function whose domain can be partitioned into finitely many pieces on which the function coincides with a polynomial. Given a set of finitely many polynomials, we show that there are only finitely many continuous selections of it and each one is semi-algebraic. Then, we establish some generic properties regarding the critical points, defined by the Clarke subdifferential, of these continuous selections. In particular, given a set of finitely many polynomials with generic coefficients, we show that the critical points of all continuous selections of it are finite and the critical values are all different, and we also derive the coercivity of those continuous selections which are bounded from below. We point out that some existing results about Lojasiewicz s inequality and error bounds for the maximum function of some finitely many polynomials can be extended to all the continuous selections of them.