Combinatorial optimization M.Tech subject

1. Combinatorial Optimization
CS-724
Lec-2: Date: 23-02-2021
Dr. Parikshit Saikia
Assistant Professor
Department of Computer Science and Engineering
NIT Hamirpur (HP)
India

2. Example 2 : Travelling Salesman Problem
We are given an integer n > 0 and the distance between every pair of n
cities in the form of an n x n matrix [di,j], where di,j Z+ . A tour is a
close path that visits every city exactly once. The problem is to find a
tour with minimum total length.
F= {all cyclic permutations π on n cities}
C(π) → σ𝒋 𝒏 𝒅i,j Here city j is visited after visiting city i
How many possible tours?

4. Local and Global Optima
Local optima
It is the extrema (minimum or maximum) of the objective
function for a given region of the input space.
• Given an instance (F, c) of an optimization problem and a neighborhood N,
a feasible solution f is called locally optimal with respect to N if
c(f) ≤ c(g) for all g N(f)
Here N(f) is defined as a set of points that are close in some
sense to the point f.
Global Optima:
It is the extrema (minimum or maximum) of the objective function for
the entire input space

7. 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

9. Properties of Convex Set
1. The intersection of ∩i I Ci of any collection {Ci | i I } of convex
set is convex.
2. The vector sum C1 + C2 of two convex set C1 and C2 is convex.
3. The set αC is convex for any convex set C and scalar α.