Quasi-Convex and Pseudo-Convex Functions on Solid Convex Sets.
STANFORD UNIV CALIF OPERATIONS RESEARCH HOUSE
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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 Psix 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 Phix 12x,Dx c,x. Analogous results hold under the additional condition that the set aDac 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
- Theoretical Mathematics
- Operations Research