An increasing function f : R n → R ∪ +∞ is a function such that f(x′) ≥ f(x) whenever x′ ≥ x (component-wise). A downward set G ⊂ R n is a set such that x ∈ G whenever x′ ≥ x for some x′ ∈ G. We present a geometric theory of monotonicity in which increasing functions relate to downward sets in the same way as convex functions relate to convex sets. By giving a central role to a separation property of downward sets similar to that of convex sets, a theory of monotonic optimization can be developed which parallels d.c. optimization in several respects.
- Downward sets
- Normal sets
- Separation property
- Increasing functions
- Monotonic functions
- Difference of monotonic functions (d.m. functions)
- Abstract convex analysis
- Global optimization
This research has been supported in part by the VN National Program on Basic Research
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Tuy, H. (2005). Monotonicity in the Framework of Generalized Convexity. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds) Generalized Convexity, Generalized Monotonicity and Applications. Nonconvex Optimization and Its Applications, vol 77. Springer, Boston, MA. https://doi.org/10.1007/0-387-23639-2_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-23638-4
Online ISBN: 978-0-387-23639-1