![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?](https://image.slidesharecdn.com/co-lec-2-210526055639/85/Combinatorial-optimization-CO-2-2-320.jpg)
![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-2-210526055639/85/Combinatorial-optimization-CO-2-7-320.jpg)
Combinatorial optimization CO-2
- 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.