A generalization of a theorem of Arrow, Barankin and Blackwell to a nonconvex case
- Authors: Kasimbeyli, Nergiz , Kasimbeyli, Refail , Mammadov, Musa
- Date: 2016
- Type: Text , Journal article
- Relation: Optimization Vol. 65, no. 5 (May 2016), p. 937-945
- Full Text:
- Reviewed:
- Description: The paper presents a generalization of a known density theorem of Arrow, Barankin, and Blackwell for properly efficient points defined as support points of sets with respect to monotonically increasing sublinear functions. This result is shown to hold for nonconvex sets of a partially ordered reflexive Banach space.
Optimality conditions in nonconvex optimization via weak subdifferentials
- Authors: Kasimbeyli, Refail , Mammadov, Musa
- Date: 2011
- Type: Text , Journal article
- Relation: Nonlinear Analysis, Theory, Methods and Applications Vol. 74, no. 7 (2011), p. 2534-2547
- Full Text:
- Reviewed:
- Description: In this paper we study optimality conditions for optimization problems described by a special class of directionally differentiable functions. The well-known necessary and sufficient optimality condition of nonsmooth convex optimization, given in the form of variational inequality, is generalized to the nonconvex case by using the notion of weak subdifferentials. The equivalent formulation of this condition in terms of weak subdifferentials and augmented normal cones is also presented. © 2011 Elsevier Ltd. All rights reserved.
On weak subdifferentials, directional derivatives, and radial epiderivatives for nonconvex functions
- Authors: Kasimbeyli, Refail , Mammadov, Musa
- Date: 2009
- Type: Text , Journal article
- Relation: Siam Journal on Optimization Vol. 20, no. 2 (2009), p. 841-855
- Full Text:
- Reviewed:
- Description: In this paper we study relations between the directional derivatives, the weak subdifferentials, and the radial epiderivatives for nonconvex real-valued functions. We generalize the well-known theorem that represents the directional derivative of a convex function as a pointwise maximum of its subgradients for the nonconvex case. Using the notion of the weak subgradient, we establish conditions that guarantee equality of the directional derivative to the pointwise supremum of weak subgradients of a nonconvex real-valued function. A similar representation is also established for the radial epiderivative of a nonconvex function. Finally the equality between the directional derivatives and the radial epiderivatives for a nonconvex function is proved. An analogue of the well-known theorem on necessary and sufficient conditions for optimality is drawn without any convexity assumptions.