{"title":"鞍点动力学下鞍点的渐近稳定性","authors":"A. Cherukuri, J. Cortés","doi":"10.1109/ACC.2015.7171030","DOIUrl":null,"url":null,"abstract":"This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient-descent in the first variable and gradient-ascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics and the function along the proximal normals to the saddle set. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.","PeriodicalId":223665,"journal":{"name":"2015 American Control Conference (ACC)","volume":"16 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2015-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"12","resultStr":"{\"title\":\"Asymptotic stability of saddle points under the saddle-point dynamics\",\"authors\":\"A. Cherukuri, J. Cortés\",\"doi\":\"10.1109/ACC.2015.7171030\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient-descent in the first variable and gradient-ascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics and the function along the proximal normals to the saddle set. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.\",\"PeriodicalId\":223665,\"journal\":{\"name\":\"2015 American Control Conference (ACC)\",\"volume\":\"16 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2015-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"12\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2015 American Control Conference (ACC)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/ACC.2015.7171030\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2015 American Control Conference (ACC)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ACC.2015.7171030","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Asymptotic stability of saddle points under the saddle-point dynamics
This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient-descent in the first variable and gradient-ascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics and the function along the proximal normals to the saddle set. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.