![Convex Set
• A set S ⊆ Rn is called convex if the line segment joining any two points
is in S. Mathematically for all x, y Rn and λ [0, 1]
z = λx + (1- λ)y S
Convex Set Non Convex Set](https://image.slidesharecdn.com/co-lec-3-210526055640/85/Combinatorial-optimization-CO-3-2-320.jpg)
![ Let S ⊆ Rn be a convex set. The function
c: S R1
is convex in S if for any two points x, y S
c(λx + (1- λ)y) ≤ λc(x) + (1-λ)c(y)
λ R1 and λ [0, 1]
If S = Rn, we simply say that c is convex.](https://image.slidesharecdn.com/co-lec-3-210526055640/85/Combinatorial-optimization-CO-3-7-320.jpg)
Combinatorial optimization CO-3
- 1. Combinatorial Optimization CS-724 Lec-3: Date: 24-02-2021 Dr. Parikshit Saikia Assistant Professor Department of Computer Science and Engineering NIT Hamirpur (HP) India
- 2. Convex Set • A set S ⊆ Rn is called convex if the line segment joining any two points is in S. Mathematically for all x, y Rn and λ [0, 1] z = λx + (1- λ)y S Convex Set Non Convex Set
- 3. Example 1: The entire set Rn is convex. Example 2: In R1, any interval is convex and any convex set is interval. Interval 1 Interval 2
- 4. Example 3: In R2, convex set loosely speaking, are those without indentations.
- 5. Convex subsets of Euclidean plane (R2)
- 6. Archimedean solid: From left Truncatedtetrahedron, cuboctahedron and truncated icosidodecahedron, and Rhombicuboctahedron Platonic solid: From left tetrahedron, cube and octahedron, and Dodecahedron In Euclidean 3 dimensional space
- 7. Let S ⊆ Rn be a convex set. The function c: S R1 is convex in S if for any two points x, y S c(λx + (1- λ)y) ≤ λc(x) + (1-λ)c(y) λ R1 and λ [0, 1] If S = Rn, we simply say that c is convex.
- 9. A function (in black) is convex if and only if the region above its graph (in green) is a convex set. This region is the function's epigraph. The epigraph or supergraph of a function f: X -> R Ս {±∞} is the set of points lying above its graph A function is convex if and only if its epigraph is a convex set.
- 12. Is every linear function convex? Let f: R R. Is f(x) = x2 Convex? Example of Convex Function in Rn ? What is Concave functions?