A Metaheuristic Framework for Bi-level Programming Problems with Multi-disciplinary Applications

Bi-level programming problems arise in situations when the decision maker has to take into account the responses of the users to his decisions. Several problems arising in engineering and economics can be cast within the bi-level programming framework. The bi-level programming model is also known as a Stackleberg or leader-follower game in which the leader chooses his variables so as to optimise his objective function, taking into account the response of the follower(s) who separately optimise their own objectives, treating the leader’s decisions as exogenous. In this chapter, we present a unified framework fully consistent with the Stackleberg paradigm of bi-level programming that allows for the integration of meta-heuristic algorithms with traditional gradient based optimisation algorithms for the solution of bi-level programming problems. In particular we employ Differential Evolution as the main meta-heuristic in our proposal.We subsequently apply the proposed method (DEBLP) to a range of problems from many fields such as transportation systems management, parameter estimation and game theory. It is demonstrated that DEBLP is a robust and powerful search heuristic for this class of problems characterised by non smoothness and non convexity

Existence of Nash Equilibria in Selfish Routing Problems

The problem of routing traffic through a congested network is studied. The framework is that introduced by Koutsoupias and Papadimitriou where the network is constituted by m parallel links, each having a finite capacity, and there are n selfish (noncooperative) agents wishing to route their traffic through one of these links: thus the problem sets naturally in the context of noncooperative games. Given the lack of coordination among the agents in large networks, much effort has been lavished in the framework of mixed Nash equilibria where the agent’s routing choices are regulated by probability distributions, one for each agent, which let the system reach thus a stochastic steady state from which no agent is willing to unilaterally deviate. Recently Mavronicolas and Spirakis have investigated fully mixed equilibria, where agents have all non zero probabilities to route their traffics on the links. In this work we concentrate on constrained situations where some agents are forbidden (have probability zero) to route their traffic on some links: in this case we show that at most one Nash equilibrium may exist and we give necessary and sufficient conditions on its existence; the conditions relating the traffic load of the agents. We also study a dynamic behaviour of the network, establishing under which conditions the network is still in equilibrium when some of the constraints are removed. Although this paper covers only some specific subclasses of the general problem, the conditions found are all effective in the sense that given a set of yes/no routing constraints on each link for each agent, we provide the probability distributions that achieve the unique Nash equilibrium associated to the constraints (if it exists)

Equilibrium Assignments in Competitive and Cooperative Traffic Flow Routing

Part 18: Optimization in Collaborative NetworksInternational audienceThe goal of the paper is to demonstrate possibilities of collaborative transportation network to minimize total travel time of the network users. Cooperative and competitive traffic flow assignment systems in case of m ≥2 navigation providers (Navigators) are compared. Each Navigator provides travel guidance for its customers (users) on the non-general topology network of parallel links. In both cases the main goals of Navigators are to minimize travel time of their users but the behavioral strategies are different. In competitive case the behavioral strategy of each Navigator is to minimize travel time of traffic flow of its navigation service users while in cooperative case – to minimize travel time of overall traffic flow. Competitive routing is formalized mathematically as a non-zero sum game and cooperative routing is formulated as an optimization problem. It is demonstrated that Nash equilibrium in the navigation game appears to be not Pareto optimal. Eventually it is shown that cooperative routing systems in smart transportation networked environments could give users less value of travel time than competitive one

Convergence time to nash equilibria

Abstract. We study the number of steps required to reach a pure Nash Equilibrium in a load balancing scenario where each job behaves selfishly and attempts to migrate to a machine which will minimize its cost. We consider a variety of load balancing models, including identical, restricted, related and unrelated machines. Our results have a crucial dependence on the weights assigned to jobs. We consider arbitrary weights, integer weights, K distinct weights and identical (unit) weights. We look both at an arbitrary schedule (where the only restriction is that a job migrates to a machine which lowers its cost) and specific efficient schedulers (such as allowing the largest weight job to move first).