{"title":"具有椭球不确定性集的不确定lcp间隙函数公式$$\\Gamma $$ -鲁棒对应物解的存在性","authors":"Lulin Tan, Wei Hong Yang, Jinbiao Pan","doi":"10.1007/s10898-023-01340-6","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we give some existence theorems of solutions to <span>\\(\\Gamma \\)</span>-robust counterparts of gap function formulations of uncertain linear complementarity problems, in which <span>\\(\\Gamma \\)</span> plays a role in adjusting the robustness of the model against the level of conservatism of solutions. If the <span>\\(\\Gamma \\)</span>-robust uncertainty set is nonconvex, it is hard to prove the existence of solutions to the corresponding robust counterpart. Using techniques of asymptotic functions, we establish existence theorems of solutions to the corresponding robust counterpart. For the case of nonconvex <span>\\(\\Gamma \\)</span>-robust ellipsoidal uncertainty sets, these existence results are not proved in the paper [Krebs et al., Int. Trans. Oper. Res. 29 (2022), pp. 417–441]; for the case of convex <span>\\(\\Gamma \\)</span>-robust ellipsoidal uncertainty sets, our existence theorems are obtained under the conditions which are much weaker than those in Krebs’ paper. Finally, a case study for the uncertain traffic equilibrium problem is considered to illustrate the effects of nonconvex uncertainty sets on the level of conservatism of robust solutions.\n</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence of solutions to $$\\\\Gamma $$ -robust counterparts of gap function formulations of uncertain LCPs with ellipsoidal uncertainty sets\",\"authors\":\"Lulin Tan, Wei Hong Yang, Jinbiao Pan\",\"doi\":\"10.1007/s10898-023-01340-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we give some existence theorems of solutions to <span>\\\\(\\\\Gamma \\\\)</span>-robust counterparts of gap function formulations of uncertain linear complementarity problems, in which <span>\\\\(\\\\Gamma \\\\)</span> plays a role in adjusting the robustness of the model against the level of conservatism of solutions. If the <span>\\\\(\\\\Gamma \\\\)</span>-robust uncertainty set is nonconvex, it is hard to prove the existence of solutions to the corresponding robust counterpart. Using techniques of asymptotic functions, we establish existence theorems of solutions to the corresponding robust counterpart. For the case of nonconvex <span>\\\\(\\\\Gamma \\\\)</span>-robust ellipsoidal uncertainty sets, these existence results are not proved in the paper [Krebs et al., Int. Trans. Oper. Res. 29 (2022), pp. 417–441]; for the case of convex <span>\\\\(\\\\Gamma \\\\)</span>-robust ellipsoidal uncertainty sets, our existence theorems are obtained under the conditions which are much weaker than those in Krebs’ paper. Finally, a case study for the uncertain traffic equilibrium problem is considered to illustrate the effects of nonconvex uncertainty sets on the level of conservatism of robust solutions.\\n</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2023-11-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10898-023-01340-6\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10898-023-01340-6","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Existence of solutions to $$\Gamma $$ -robust counterparts of gap function formulations of uncertain LCPs with ellipsoidal uncertainty sets
In this paper, we give some existence theorems of solutions to \(\Gamma \)-robust counterparts of gap function formulations of uncertain linear complementarity problems, in which \(\Gamma \) plays a role in adjusting the robustness of the model against the level of conservatism of solutions. If the \(\Gamma \)-robust uncertainty set is nonconvex, it is hard to prove the existence of solutions to the corresponding robust counterpart. Using techniques of asymptotic functions, we establish existence theorems of solutions to the corresponding robust counterpart. For the case of nonconvex \(\Gamma \)-robust ellipsoidal uncertainty sets, these existence results are not proved in the paper [Krebs et al., Int. Trans. Oper. Res. 29 (2022), pp. 417–441]; for the case of convex \(\Gamma \)-robust ellipsoidal uncertainty sets, our existence theorems are obtained under the conditions which are much weaker than those in Krebs’ paper. Finally, a case study for the uncertain traffic equilibrium problem is considered to illustrate the effects of nonconvex uncertainty sets on the level of conservatism of robust solutions.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
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