On directions of strictness, affinity and constancy and the minimum set of a proper convex function
- 1 January 1988
- journal article
- research article
- Published by Taylor & Francis in Optimization
- Vol. 19 (2) , 157-167
- https://doi.org/10.1080/02331938808843331
Abstract
The common definition of directions of affinity resp. constancy of a proper convex function implicitly requires the domain of finiteness to be unbounded in these directions. This assumption is dropped here, and directions of strictness are also defined. Under a mild continuity condition it is shown that directions of affinity resp. constancy form vector spaces. Behaviour under addition and linear transformations is investigated. Particularly simple results are obtained under the assumption that any direction is either a direction of strictness or affinity. These results are used to obtain a theorem on the dimension of the minimum set of a proper convex function, as well as simple rank conditions necessary and sufficient for strict convexity of a sum, where the summands are not necessarily strictly convex. The linear Poisson model and geometric programming are treated as examples.Keywords
This publication has 5 references indexed in Scilit:
- Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in Generalized Linear ModelsThe Annals of Statistics, 1985
- Conjugate Direction Methods in OptimizationPublished by Springer Nature ,1980
- On the existence and uniqueness of the maximum likelihood estimates for certain generalized linear modelsBiometrika, 1976
- Convex AnalysisPublished by Walter de Gruyter GmbH ,1970
- Convexity and Optimization in Finite Dimensions IPublished by Springer Nature ,1970