Accession Number : AD0724751

Title :   Quasi-Convex and Pseudo-Convex 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 quasi-convexity (pseudo-convexity) of quadratic functions on solid convex sets can be reduced to an examination of finitely many conditions. One determines two maximal domains of quasi-convexity (pseudo-convexity) 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 quasi-convex (pseudo-convex) 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 (semi-positive) 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 quasi-convex objective function and constraint functions. Finally, one gives a necessary condition and a sufficient condition for the quasi-convexity of a function in Class C squared (i.e., twice continuously differentiable) on a solid convex set. One also establishes a relation between the quasi-convexity and the pseudo-convexity 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