Combinatorial optimization M.Tech subject

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?