Approximate solution of the HJB inequality related to the infinite horizon optimal control problem with discounting

This paper is focusing on finding smooth approximate solutions of the HJB inequality that corresponds to the infinite horizon optimal control problem with discounting. We establish that such approximate solutions exist (under a simple controllability type condition) and that they can be used for construction of near optimal controls. We also show that these approximate solutions of the HJB inequality can be found by solving certain semi-infinite linear programming problems and we propose an algorithm for the solution of the latter. We discuss a numerical solution of a non-trivial optimal control problem obtained with the help of a software implementation of the new algorithm.28 page(s

A note on using the resistance-distance matrix to solve Hamiltonian cycle problem

An instance of Hamiltonian cycle problem can be solved by converting it to an instance of Travelling salesman problem, assigning any choice of weights to edges of the underlying graph. In this note we demonstrate that, for difficult instances, choosing the edge weights to be the resistance distance between its two incident vertices is often a good choice. We also demonstrate that arguably stronger performance arises from using the inverse of the resistance distance. Examples are provided demonstrating benefits gained from these choices