
Accession Number : AD0724751
Title : QuasiConvex and PseudoConvex Functions on Solid Convex Sets.
Descriptive Note : Technical rept.,
Corporate Author : STANFORD UNIV CALIF OPERATIONS RESEARCH HOUSE
Personal Author(s) : Ferland,Jacques A.
Report Date : APR 1971
Pagination or Media Count : 77
Abstract : The purpose of the paper is to prove that testing quasiconvexity (pseudoconvexity) of quadratic functions on solid convex sets can be reduced to an examination of finitely many conditions. One determines two maximal domains of quasiconvexity (pseudoconvexity) for the quadratic form Psi(x) = (x,Dx) where D has exactly one negative eigenvalue, and conversely, one shows that if the quadratic form Psi is quasiconvex (pseudoconvex) on a solid convex set, then the matrix D has exactly one negative eignevalue and the solid convex set is contained in one of the maximal domains. The special case when the solid convex set is the nonnegative (semipositive) orthant is also analyzed. This study is then extended to quadratic functions Phi(x) = 1/2(x,Dx) + (c,x). Analogous results hold under the additional condition that the set (a/Da+c = 0) is not empty. In the last part of this paper, one analyzes functions that are not necessarily quadratic. One obtains some results on mathematical programming problems having twice differentiable quasiconvex objective function and constraint functions. Finally, one gives a necessary condition and a sufficient condition for the quasiconvexity of a function in Class C squared (i.e., twice continuously differentiable) on a solid convex set. One also establishes a relation between the quasiconvexity and the pseudoconvexity of twice differentiable functions on solid convex sets. (Author)
Descriptors : (*MATHEMATICAL PROGRAMMING, *CONVEX SETS), FUNCTIONS(MATHEMATICS), MATRICES(MATHEMATICS), INEQUALITIES, PARTIAL DIFFERENTIAL EQUATIONS, QUADRATIC PROGRAMMING, THEOREMS
Subject Categories : Theoretical Mathematics
Operations Research
Distribution Statement : APPROVED FOR PUBLIC RELEASE