Classic Lagrangian may not be applicable to the traveling salesman problem

TL;DR

This paper investigates the limitations of using the classic Lagrangian approach for the traveling salesman problem, demonstrating through a specific example that it may not be applicable.

Contribution

It constructs the dual problem for TSP via the classic Lagrangian and shows its limitations through a numerical example.

Findings

Classic Lagrangian may not be suitable for TSP

Optimality conditions can be expressed as an inverse problem

Numerical example with 4 cities illustrates the limitation

Abstract

In this short note, the dual problem for the traveling salesman problem is constructed through the classic Lagrangian. The existence of optimality conditions is expressed as a corresponding inverse problem. A general 4-cities instance is given, and the numerical experiment shows that the classic Lagrangian may not be applicable to the traveling salesman problem.