A two-level value function is an optimal value function of a parametric optimization problem where the feasible set is described by the optimal solution set of another optimization problem. The primary goal of this project is to develop, for the first time ever in the literature, explicit approximations of two-level value functions. Being able to approximate these functions will enable the design of simple and efficient algorithms for unsolved problems in various areas of optimization, including multilevel, robust, and stochastic optimization. As pessimistic bilevel optimization represents the most prominent application of two-level value functions, the efficiency of the approximations developed in this project will be evaluated though algorithms to be constructed to solve the pessimistic bilevel program. This is one of the most challenging problems in the field of optimization, as the objective function does not have an explicit analytical expression and is typically only upper semicontinuous. These features of pessimistic bilevel optimizaion place the problem out of the framework of standard optimization, where the objective function to be minimized is usually given explicitly and required to be at least lower semicontinuous. Solving the pessimistic bilevel program will unlock the potential of bilevel optimization as a powerful tool for optimal decision-making. To see this, note that the most basic rule in a bilevel optimization problem (also known as Stackelberg game) is that the leader (upper-level player) plays first by selecting a decision value that optimizes his/her utility function. Subsequently, the follower (lower-level player) selfishly reacts to this choice from the leader by choosing his/her own decision value that optimizes his/her utility function. This generally gives rise to two scenarios: the optimistic and pessimistic bilevel programs. In the optimistic case, one assumes that the follower will cooperate to make decisions that are in favour of the leader. However, if the leader is uncertain about the cooperation of the follower, as a risk-averse player, he/she will solve the pessimistic bilevel program to minimize any potential damage that might result from unfavorable choices from the follower. It is therefore clear that most practical applications of bilevel optimization will only fit into the pessimistic model, as it is more realistic for the leader to assume that the follower will not play in his/her favour. However, because solving the pessimistic bilevel program is very difficult, the literature on bilevel optimization has almost ignored the problem, and is therefore essentially concentrated around the optimistic model of the problem. This project will shift focus from optimistic to pessimistic bilevel optimization, while creating the first framework to efficiently solve the problem. More broadly, bilevel optimization represents one of the most popular problems in the field of optimization thanks to its inherent mathematical challenges, as well as the wide range of applications which have been growing exponentially in the last 40 years. The results from this project can help to solve problems of major importance in the UK and internationally. For instance, for the large transportation projects currently planned or ongoing in the UK (e.g., Crossrail, HS2, and Heathrow 3rd Runway), bilevel optimization offers a framework for the government (as upper-level player) to maximize their outputs while ensuring that taxpayers (as lower-level player) are also able to achieve their expected objectives. Suitable bilevel optimization models in this context can be constructed around the optimal network design, establishing optimal toll policies where necessary or the optimal estimation of the demand for these facilities.

An efficient transportation system is vital to the economic and social well-being of large cities. The transport demand implied by economic growth, however, requires transport networks to become more and more complex, making their management difficult. Fortunately, modern systems such as the London Underground generate vast amounts of data that can be analysed to better understand passenger behaviour and needs. Besides understanding the typical daily patterns that we can observe on a regular basis, Data Science methods allows us to look into in the less usual events such as unplanned disruptions that are still important to any user, and to also model individualised behaviour instead of only aggregates. In a large system such as the London Underground, signal failures and disruptive events eventually take place, requiring passengers to change plans in a variety of ways. This research provides advanced statistical modelling and machine learning approaches to learn from past events to examine how passengers adapt themselves when a disruption occurs. When a disruption takes place, the model will provide information of likely changes, such as increased number of passengers leaving a station because they could not reach their destination. These models are important for transport authorities to understand the resilience of the system, different combinations of location and time of a disruption, and unusual responses from passengers that may motivate different communication strategies to inform users of better travel adjustments. This research also opens up conceptual ideas to be exploited in the future using new technologies to monitor and adaptively respond to passenger needs in a more optimised and time-effective way.

Path choices by transport network users depend in part on decisions made in response to foreseeable circumstances at the start of the trip and in part on choices made as the trip unfolds. In the case of transit networks, on-trip choices may depend on factors such as which line arrives first at a particular stop or platform, whereas in the case of traffic networks, on-trip choices may be influenced by factors such as an encountered queue, the state of traffic signals, or information received by the driver. In transit assignment, the effects of on-trip decisions are accommodated through the definition of a set of elemental paths that may be optimal (referred to as a hyperpath) and strategies for on-trip choices. In traffic assignment, the stochastic user equilibrium principle has been used to capture the effects of on-trip choices and different user preferences. However, the usual forms of stochastic assignment (C-logit, path size logit, cross-nested logit or probit) generally place few or ad hoc constraints on the set of feasible paths and assume that all feasible paths have a non-zero probability of use. Choice of path is assumed to be the result of random utility variation rather than the outcome of a choice strategy in the face of unfolding circumstances as the trip takes place. To this extent, the treatment of route choice behaviour is more sophisticated in transit than in traffic assignment. Where congestion and service disruptions are a prominent feature, it is necessary to take time-dependency into account in defining hyperpaths. The objective of this proposal is to extend the concepts of hyperpath and strategy-based decision-making to dynamic transit networks, traffic networks and multi-modal networks and then assess the benefits of so doing.

We propose a Research Cluster to explore the opportunities and challenges of the Digital Economy. The Internet is driving many powerful convergences in media, devices and infrastructure provision. These convergences hold the promise that the next-generation Internet could be a very powerful and universal platform where a great deal of economic and social activity could take place. Given the universal nature of the Internet this platform would break down the traditional distinctions between, say business and the general public, and anyone from any sector could be a provider or a user of these services.If properly realised the benefits from these developments could be considerable. However, they will not happen automatically, there are many issues that need to be tackled before they can be fully achieved. Given the nature of the Internet these issues are as much economic, social, legal and regulatory as they are technical and, critically, these issues have been tackled together to provide an holistic and complete solution.We have assembled a multi-disciplinary consortium that includes talented and experienced research workers in all the fields necessary to address these issues and have established relationships with major stakeholders in the next-generation Internet. The Cluster is led by Imperial College, the University of Oxford and the University of Southampton. The Research Cluster would conduct an open investigation to identify the topics that need to be addressed and produce a roadmap or research and development agenda to tackle them. The Cluster proposes to hold two open workshops at the beginning and end of the one year study to involve the community as much as possible and to create expert Working Groups to address the critical issues. All these deliberations will be conducted in an open manner using Web 2.0 community networking techniques.The outputs of these deliberations will a programme of linked actions to drive forward the development of the Digital Economy. These will comprise multi-disciplinary research programmes, commercial exploitations, social or legal actions or regulatory recommendations. The Reserch Cluster will also be used to identify the coalitions, again both research and commercial, best suited to take forward these proposals.