an:05709190
Zbl 1231.90314
Zhao, Y. X.; Wang, S. Y.; Coladas Uria, L.
Characterizations of \(r\)-convex functions
EN
J. Optim. Theory Appl. 145, No. 1, 186-195 (2010).
00261083
2010
j
90C25
strict \(r\)-convexity; \(r\)-convexity; positive-semidefinite matrices; positive-definite matrices; convex functions; quasiconvex functions
Summary: This paper discusses some properties of \(r\)-convexity and its relations with some other types of convexity. A characterization of convex functions in terms of \(r\)-convexity is given without assuming differentiability. The concept of strict \(r\)-convexity is introduced. For a twice continuously differentiable function \(f\), it is shown that the strict \(r\)-convexity of \(f\) is equivalent to a certain condition on \(\nabla ^{2} f\). Further, it is shown that this condition is satisfied by quasiconvex functions satisfying a less stringent condition.