{"title":"二次型最小值的计算(初步报告)","authors":"A. Yao","doi":"10.1145/800116.803749","DOIUrl":null,"url":null,"abstract":"The following problem was recently raised by C. William Gear [1]: Let F(x1,x2,...,xn) = &Sgr;i≤j a'ijxixj + &Sgr;i bixi +c be a quadratic form in n variables. We wish to compute the point x→(0) = (x1(0),...,xn(0)), at which F achieves its minimum, by a series of adaptive functional evaluations. It is clear that, by evaluating F(x→) at 1/2(n+1)(n+2)+1 points, we can determine the coefficients a'ij,bi,c and thereby find the point x→(0). Gear's question is, “How many evaluations are necessary?” In this paper, we shall prove that O(n2) evaluations are necessary in the worst case for any such algorithm.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1975-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"On computing the minima of quadratic forms (Preliminary Report)\",\"authors\":\"A. Yao\",\"doi\":\"10.1145/800116.803749\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The following problem was recently raised by C. William Gear [1]: Let F(x1,x2,...,xn) = &Sgr;i≤j a'ijxixj + &Sgr;i bixi +c be a quadratic form in n variables. We wish to compute the point x→(0) = (x1(0),...,xn(0)), at which F achieves its minimum, by a series of adaptive functional evaluations. It is clear that, by evaluating F(x→) at 1/2(n+1)(n+2)+1 points, we can determine the coefficients a'ij,bi,c and thereby find the point x→(0). Gear's question is, “How many evaluations are necessary?” In this paper, we shall prove that O(n2) evaluations are necessary in the worst case for any such algorithm.\",\"PeriodicalId\":20566,\"journal\":{\"name\":\"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1975-05-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/800116.803749\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/800116.803749","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On computing the minima of quadratic forms (Preliminary Report)
The following problem was recently raised by C. William Gear [1]: Let F(x1,x2,...,xn) = &Sgr;i≤j a'ijxixj + &Sgr;i bixi +c be a quadratic form in n variables. We wish to compute the point x→(0) = (x1(0),...,xn(0)), at which F achieves its minimum, by a series of adaptive functional evaluations. It is clear that, by evaluating F(x→) at 1/2(n+1)(n+2)+1 points, we can determine the coefficients a'ij,bi,c and thereby find the point x→(0). Gear's question is, “How many evaluations are necessary?” In this paper, we shall prove that O(n2) evaluations are necessary in the worst case for any such algorithm.
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