Robust bi-level optimization models in transportation science

Abstract

Mathematical programmes with equilibrium constraints (MPECs) constitute important modelling tools for network flow problems, as they place 'what-if' analyses in a proper mathematical framework. We consider a class of stochastic MPEC traffic models that explicitly incorporate possible uncertainties in travel costs and demands. In stochastic programming terminology, we consider 'here-and-now' models where decisions must be made before observing the uncertain parameter values and the responses of the network users; the objective is to minimize the expectation of the upper-level objective function. Such a model could, for example, be used to derive a fixed toll pricing scheme that provides the best revenue for a given network over a time period, where variations in traffic conditions and demand elasticities are described by distributions of parameters in the travel time and demand functions.We present new results on the stability of globally optimal solutions to perturbations in the probability distribution, establishing the robustness of the model. We also discuss penalization and discretization algorithms, the latter enabling the use of standard MPEC algorithms, and provide many future research avenues.

On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 2: Applications

“…Let val(P) denote the optimal value of problem P. The following result shows the stability of globally optimal solutions. The corresponding result in the context of topology optimization in structural mechanics can be found in [18] and to network design under traffic equilibrium in [45,46]. The proof presented here is similar.…”

“…We can, and should, therefore formulate the network design problem as an SMPEC, which gives us a design which is the best possible on average. This problem has been studied in Patriksson [45,46] and Birbil et al [6].…”

“…For the application in radiation therapy, F(x, y(ω), ω) = K(ω)x − y(ω), where we consider the influence matrix K to depend on the stochastic variable ω, and so Assumption (B2) is trivially fulfilled. For traffic equilibrium, we refer to Patriksson [45,46] for details on the mapping F. Furthermore, conditions (B3) and (B4) are natural to assume for the two applications.…”

Transport link scanner: simulating geographic transport network expansion through individual investments

“…In the 1960s, conceptual and empirical modelling efforts have been undertaken by quantitative geographers (Taaffe et al 1963;Warntz 1966;Kolars and Malin 1970). More recently, network optimality and bi-level optimization methods (Patriksson 2008;Youn et al 2008;Li et al 2010), the role of selforganization and the role of ownership (Xie and Levinson 2007) have been investigated in controlled conditions. This has been followed by empirically based exercises to test heuristic network design optimization methods (Vitins and Axhausen 2009), to understand the driving forces of network growth (Rietveld and Bruinsma 1998) and the role of first mover advantages and to forecast future network investments in a fairly mature transport system (Levinson et al 2012).…”

Decision Rule Bounds for Two-Stage Stochastic Bilevel Programs

“…Despite their rich modeling power, stochastic bilevel programs have attracted much less attention-presumably because they are perceived as substantially harder than their (already difficult) deterministic counterparts. Optimistic stochastic bilevel models emerged in the context of truss topology optimization [7,29], traffic planning [1,28], network design [2] and strategic pricing in electricity markets [16] etc. We note that these optimistic models can also be viewed as special instances of stochastic mathematical programs with equilibrium constraints [14,30,33,41,42].…”