147. Curve Based Convex Functions and its Optimality
Invited abstract in session MC-9: Generalized convexity and monotonicity 2, stream Generalized convexity and monotonicity.
Monday, 14:00-16:00Room: B100/8013
Authors (first author is the speaker)
| 1. | Qamrul Hasan Ansari
|
| Mathematics, King Fahd University of Petroleum and Minerals |
Abstract
This paper introduces a new concept of convexity, namely α-convexity or curve-based convexity. It is found that an α-convex set can be disconnected—thereby making the proposed α-convexity completely different than geodesic convexity since a geodesically convex set must be path-connected. We discuss continuity, differentiability, a direction of descent, epigraph, and extremum properties of α-convex functions. It is found that a curve-based convex function may not have directional derivative. Necessary and sufficient optimality conditions of first and second orders for unconstrained optimization, and also a sufficient Karush-Kuhn-Tucker type condition for constrained optimization problems with α-convex functions are established. Further, we have proposed the concept of bifunction along a curve and discuss the nonsmooth variational inequality problems in terms of α-bifunctions. Finally, a relation providing a connection between the gap function and α-bifunctions for variational inequality problems is given. This work will allow the use of standard optimization methods and pave the road to various extensions of conventional optimization tools.
Keywords
- First-order optimization
- Linear and nonlinear optimization
- Global optimization
Status: accepted
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