Mathematical Program With Equilibrium Constraints

This paper considers the optimal toll design problem that uses the Probit model to determine travellers' route-choices. Under probit, the route flow solution to the resulting stochastic user equilibrium (SUE) is unique and can be stated implicitly as a function of tolls. This reduces the toll design problem to an optimisation problem with only nonnegativity constraints. Additionally, the gradient of the objective function can be approximated using the chain rule and the first order Taylor approximation of the equilibrium condition. To determine SUE, this paper considers two techniques. One uses Monte-Carlo simulation to estimate route choice probabilities and the method of successive averages with its prescribed step length. The other relies on the Clark approximation and computes an optimal step length. Although both are effective at solving the toll design problem, numerical experiments show that the technique with the Clark approximation is more robust on a small network.

The Network Design Problem (NDP) refers to the optimization problem faced by a planner whose aim is to improve a transport network, drawing on limited resources. Though the NDP may lead to, for example, a set of tolls that maximise social welfare, often no consideration is made of the distribution of resulting benefits and costs across the population of travellers. We consider a network under probit stochastic user equilibrium (SUE) with elastic demand, disaggregated into multiple user classes with different values of time and link-specific tolls. We propose the Theil measure for quantifying equity, which can be incorporated either into the objective function or the constraints within the NDP. A sensitivity analysis of the SUE flows provides the basis for computing the Jacobian of the social welfare function and of the Theil measure. This allows gradient-based optimisation algorithms to be used in solving the NDP. Numerical examples are reported.

Selfish routing, represented by the User-Equilibrium (UE) model, is known to be inefficient when compared to the System Optimum (SO) model. However, there is currently little understanding of how the magnitude of this inefficiency, which can be measured by the Price of Anarchy (PoA), varies across different structures of demand and supply. Such understanding would be useful for both transport policy and network design, as it could help to identify circumstances in which policy interventions that are designed to induce more efficient use of a traffic network, are worth their costs of implementation. This paper identifies four mechanisms that govern how the PoA varies with travel demand in traffic networks with separable and strictly increasing cost functions. For each OD movement, these are expansions and contractions in the sets of routes that are of minimum cost under UE and minimum marginal total cost under SO. The effects of these mechanisms on the PoA are established via a combination of theoretical proofs and conjectures supported by numerical evidence. In addition, for the special case of traffic networks with BPR-like cost functions having common power, it is proven that there is a systematic relationship between link flows under UE and SO, and hence between the levels of demand at which expansions and contractions occur. For this case, numerical evidence also suggests that the PoA has power law decay for large demand.

In this paper we consider the question of solving equilibrium problems-formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPECs)-as nonlinear programs, using an interior-point approach. These problems pose theoretical difficulties for nonlinear solvers, including interior-point methods. We examine the use of penalty methods to get around these difficulties and provide substantial numerical results. We go on to show that penalty methods can resolve some problems that interior-point algorithms encounter in general.

In this work, the open loop dynamic optimisation of a large-scale natural gas processing plant is performed. A rigorous differential-algebraic equation (DAE) model has been formulated to represent main plant units, such as shell and tube heat exchangers, highpressure separator and demethanizing column. In the shell and tube heat exchangers, the hot stream partially condenses and equations to consider the partial condensation of the fluids have been included. A rigorous index one model for the demethanizing column has been developed. The DAE optimisation problem is solved with a simultaneous approach, in which both state and control variables are discretised and the original DAE optimisation model is transformed into a large-scale nonlinear problem (NLP), which is solved using Sequential Quadratic Programming (SQP) methods. Optimal profiles have been obtained for main operating variables to achieve an enhanced product recovery.

For an arbitrary finite family of semi-algebraic/definable functions, we consider the corresponding inequality constraint set and we study qualification conditions for perturbations of this set. In particular we prove that all positive diagonal perturbations, save perhaps a finite number of them, ensure that any point within the feasible set satisfies Mangasarian-Fromovitz constraint qualification. Using the Milnor-Thom theorem, we provide a bound for the number of singular perturbations when the constraints are polynomial functions. Examples show that the order of magnitude of our exponential bound is relevant. Our perturbation approach provides a simple protocol to build sequences of "regular" problems approximating an arbitrary semialgebraic/definable problem. Applications to sequential quadratic programming methods and sum of squares relaxation are provided.

Chance constraints are a valuable tool for the design of safe decisions in uncertain environments; they are used to model satisfaction of a constraint with a target probability. However, because of possible non-convexity and non-smoothness, optimizing over a chance constrained set is challenging. In this paper, we establish an exact reformulation of chance constrained problems as a bilevel problems with convex lower-levels. We then derive a tractable penalty approach, where the penalized objective is a difference-ofconvex function that we minimize with a suitable bundle algorithm. We release an easy-to-use open-source python toolbox implementing the approach, with a special emphasis on fast computational subroutines.

Chance constraints are a valuable tool for the design of safe decisions in uncertain environments; they are used to model satisfaction of a constraint with a target probability. However, because of possible non-convexity and non-smoothness, optimizing over a chance constrained set is challenging. In this paper, we establish an exact reformulation of chance constrained problems as a bilevel problems with convex lower-levels. We then derive a tractable penalty approach, where the penalized objective is a difference-ofconvex function that we minimize with a suitable bundle algorithm. We release an easy-to-use open-source python toolbox implementing the approach, with a special emphasis on fast computational subroutines.

Deregulated electricity markets use an auction mechanism to select offers and their power levels for energy and ancillary services. A settlement mechanism is then used to determine the payments resulting from the selected offers. Currently, most independent system operators (ISOs) in the United States use an auction mechanism that minimizes the total offer costs but determine payment costs using a settlement mechanism that pays uniform market clearing prices (MCPs) to all selected offers. Under this setup, the auction and settlement mechanisms are inconsistent since minimized costs are different from payment costs. Illustrative examples in the literature have shown that for a given set of offers, if an auction mechanism that directly minimizes the payment costs is used, then payment costs can be significantly reduced as compared to minimizing offer costs. This observation has led to discussions among stakeholders and policymakers in the electricity markets as to which of the two auction mechanisms is more appropriate for ISOs to use. While methods for minimizing offer costs abound, limited approaches for minimization of payment costs have been reported. This paper presents an effective method for directly minimizing payment costs. In view of the specific features of the problem including the nonseparability of its objective function, the discontinuity of offer curves, and the maximum term in defining MCPs, our key idea is to use augmented Lagrangian relaxation and to form and solve offer and MCP subproblems by using the surrogate optimization framework. Numerical testing results demonstrate that the method is effective, and the resulting payment costs are significantly lower than what are obtained by minimizing the offer costs for a given set of offers. Index Terms-Augmented Lagrangian relaxation, deregulated electricity markets, market clearing price (MCP), offer cost minimization, payment cost minimization, surrogate optimization. I. INTRODUCTION D EREGULATED wholesale electricity markets (e.g., the day-ahead, hour-ahead, and real-time markets) operated by independent system operators (ISOs) in the United States generally adopt an auction mechanism to select generation offers and demand bids and their levels for energy and ancillary services. A settlement mechanism is then used to determine the corresponding payments for the selected generation of

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problems are complicated by nonconvex agent problems and therefore providing tractable conditions for existence of global or even local equilibria has proved challenging. Consequently, much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. In the first part of the paper, we provide what is perhaps the first general existence to such games. Our central idea is to relate the global minima of certain optimization problems to equilibria of these games. We show that if the objectives of the leaders admit a quasi-potential function, one can construct an optimization problem (a mathematical program with equilibrium constraints (MPEC)) whose global minimum is an equilibrium of the game. In effect existence of equilibria can be guaranteed by the solvability of an optimization problem, which holds under mild conditions. We then consider a modified formulation in which every leader is cognizant of the equilibrium constraints of all leaders, leading to a shared constraint game. This modification allows for more general objectives functions and it is shown that when the leader objectives admit a potential function, the global minima of an optimization problem where the potential function is minimized over the shared constraint, are equilibria of the modified multi-leader multi-follower game. Importantly, equilibria of this modified game contain the equilibria, if any, of the original game. In both instances, local minima and B-stationary points of the associated optimization problem are shown to be local Nash equilibria and Nash B-stationary points of the corresponding multi-leader multi-follower game.

This note considers alternative methods for computing Wadropian (traffic network) equilibria using a multicommodity formulation in nonlinear program and complementarity formats. These methods compute exact equilibria, they are efficient and they can be implemented with standard modeling software.

The convergence of many proximal algorithms involving a gradient descent relies on its Lipschitz constant. To avoid computing it, backtracking rules can be used. While such a rule has already been designed for the forward-backward algorithm (FBwB), this scheme is not flexible enough when a non-differentiable penalization with a linear operator is added to a constraint. In this work we propose a backtracking rule for the primal-dual scheme (PDwB), and evaluate its performance for the epigraphical constrained high dynamical reconstruction in high contrast polarimetric imaging, under TV penalization.

Complementarity models can represent the simultaneous optimization problems of one or several interacting decision-makers, and thus they have become an increasingly important and powerful tool for formulating and solving bottom-up energy market models. This paper provides an overview of the full range of complementarity-based formulations and how these can be applied to assist the different market participants and organizations with their decision-making processes. To this end, the first part of the paper introduces the mathematical formulation of some basic complementarity models, which are illustrated by highly simplified but illustrative energy market applications. Considering these models, the second part of the paper is devoted to describing in broad terms four areas of their potential application: electricity markets, emission markets, natural gas markets and economies comprising several interacting markets.

The paper concerns a new class of optimization-related problems called Equilibrium Problems with Equilibrium Constraints (EPECs). One may treat them as two level hierarchical problems, which involve equilibria at both lower and upper levels. Such problems naturally appear in various applications providing an equilibrium counterpart (at the upper level) of Mathematical Programs with Equilibrium Constraints (MPECs). We develop a unified approach to both EPECs and MPECs from the viewpoint of multiobjective optimization subject to equilibrium constraints. The problems of this type are intrinsically nonsmooth and require the use of generalized differentiation for their analysis and applications. This paper presents necessary optimality conditions for EPECs in finite-dimensional spaces based an advanced generalized variational tools of variational analysis. The optimality conditions are derived in normal form under certain qualification requirements, which can be regarded as proper analogs of the classical Mangasarian-Fromovitz constraint qualification in the general settings under consideration.

Mathematical programs with equilibrium constrains (MPECs) in which the constraints are defined by a parametric variational inequality are considered. Recently, nonlinear programming solvers have been used to solve MPECs. Smoothing algorithms have been very successful. In this note, a smoothing approach based on neural network function to solve MPECs is proposed. The performance of the proposed smoothing approach on as set of well-known problems is tested. A very useful and efficient tool for practitioners to solve not only MPEC problems but also more general class of MPECs is provided.

The generalized higher order macroscopic traffic flow model is reformulated in the Lagrangian coordinate system to develop more efficient numerical method. Providing a new node model based on the Lagrangian coordinate system which is compatible with both microscopic and macroscopic descriptions. Adding discretization of node model that describe inflow and outflow boundaries to perform simulations on networks of roads. Include internal initial boundary conditions to improve the use of floating vehicle data. Node in-and out-flows calculated as asymptotic flows as a function of the demands and supplies.

This paper illustrates how a supplier profit may be affected by the market pricing mechanism under imperfect competition. A parameterized Supply Function Equilibrium (SFE) model involving manipulation of the sole intercept is used to represent the strategic behavior of each supplier. Through utilizing a bilevel optimization technique and a Mathematical Program with Equilibrium Constraints (MPECs) approach, market equilibria are calculated and compared under pay-as-bid pricing (PABP) and marginal pricing (MP) mechanisms. For an unconstrained case, analytically it is demonstrated that the optimal bidding strategy and the maximum profit of each supplier, as well as the market clearing price are the same under PABP and MP. The effects of the transmission limits and the supplier capacity constraints are discussed through numerical results.

This paper proposes a frequency based transit assignment model that accounts for online information and strict capacity constraints. A heuristic is proposed to solve the problem, which first applies an unconstrained transit assignment procedure and then handles only the overloaded transit line segments, reassigning the surplus passengers. The developed procedure is efficient, requiring very short running times compared to existing capacity constrained transit assignment models. The model assumes passengers receive online information of both predicted arrival time and occupancy condition. Two cases of occupancy information are considered: (1) passengers are informed of the vehicle occupancy, and may change their route selection accordingly, (2) passengers have no occupancy information and in case their boarding is denied, they are enforced to choose a later departing alternative. The model is applied for the Winnipeg network, and as expected the inclusion of capacity constraints increases the average travel time compared to the unconstrained model. However, prior knowledge of the occupancy condition was found to reduce the additional travel time. This result emphasizes the potential benefits of providing occupancy information to passengers.

Transportation planners and traffic engineers are faced nowadays with immense modeling challenges arising from several emerging policy, planning, and engineering developments. In fact, the recent emergence of mobile sensing and traffic monitoring technology has provided an unprecedented amount of information and data for traffic analysis, demanding the adaptation of mathematical and physical models to a new generation of cyberinfrastructure. Some of the major challenges that traffic models meet include: computational tractability, very large-scale deployment, real-time application decision support, multiple time scales, and fusion of dissimilar data. This research takes a major step in developing analytical, holistic mathematical models and traffic analysis tools capable of addressing data-enabled traffic modeling, estimation, control and optimization problems, leading to a more efficient, reliable, and sustainable transportation system. In the project, the authors constructed: (1) ...

Transportation planners and traffic engineers are faced nowadays with immense modeling challenges arising from several emerging policy, planning, and engineering developments. In fact, the recent emergence of mobile sensing and traffic monitoring technology has provided an unprecedented amount of information and data for traffic analysis, demanding the adaptation of mathematical and physical models to a new generation of cyberinfrastructure. Some of the major challenges that traffic models meet include: computational tractability, very large-scale deployment, real-time application decision support, multiple time scales, and fusion of dissimilar data. This research takes a major step in developing analytical, holistic mathematical models and traffic analysis tools capable of addressing data-enabled traffic modeling, estimation, control and optimization problems, leading to a more efficient, reliable, and sustainable transportation system. In the project, we constructed: (1) a class of Mathematical Programming with Equilibrium Constraints (MPEC) problems to understand and mitigate congestion externalities and mobile source emissions, and (2) a class of Mixed Binary Integer Programs (MBIPs) for data fusing, real-time traffic estimation and prediction, as well as data-enabled traffic control.

In this paper, we consider dynamic congestion pricing in the presence of demand uncertainty. In particular, we apply a robust optimization (RO) approach based on a bi-level cellular particle swarm optimization (BCPSO) to optimal congestion pricing problems when flows correspond to dynamic user equilibrium on the network of interest. Such a formulation is recognized as a second-best pricing problem, and we refer to it as the dynamic optimal toll problem with equilibrium constraints (DOTPEC). We then present numerical experiments in which BCPSO is compared with two alternative robust dynamic solution approaches: bi-level simulated annealing (BSA) and cutting plane-based simulated annealing (CPSA), as well as a nominal dynamic solution and a robust static solution. We show that robust dynamic solutions improve the worst case, average, and stability of total travel cost in comparison with the nominal dynamic and the robust static solutions. The numerical results also show that BCPSO outperforms BSA and CPSA in terms of solution quality and computational efficiency.

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Approximately twenty years ago the modern interest for hierarchical programming was initiated by J. Bracken and J.M. McGill [9], [10]. The activities in the field have ever grown lively, both in terms of theoretical developments and terms of the diversity of the applications. The collection of seven papers in this issue covers a diverse number of topics and provides a good picture of recent research activities in the field of bilevel and hierarchical programming. The papers can be roughly divided into three categories; Linear bilevel programming is addressed in the first two papers by Gendreau et al and Moshirvaziri et al; The following three papers by Nicholls, Loridan & Morgan, and Kalashnikov K: Kalashnikova are concerned with nonlinear bilevel programming; and, finally, Wen & Lin and Nagase 8z Aiyoshi address hierarchical decision making issues relating to both biobjective and bilevel programming.

Complementarity models can represent the simultaneous optimization problems of one or several interacting decision-makers, and thus they have become an increasingly important and powerful tool for formulating and solving bottom-up energy market models. This paper provides an overview of the full range of complementarity-based formulations and how these can be applied to assist the different market participants and organizations with their decision-making processes. To this end, the first part of the paper introduces the mathematical formulation of some basic complementarity models, which are illustrated by highly simplified but illustrative energy market applications. Considering these models, the second part of the paper is devoted to describing in broad terms four areas of their potential application: electricity markets, emission markets, natural gas markets and economies comprising several interacting markets.

This paper is concerned with adaptive signal control problems on a road network, using a link-based kinematic wave model (Han et al., 2012). Such a model employs the Lighthill-Whitham-Richards model with a triangular fundamental diagram. A variational type argument (Lax, 1957; Newell, 1993) is applied so that the system dynamics can be determined without knowledge of the traffic state in the interior of each link. A Riemann problem for the signalized junction is explicitly solved; and an optimization problem is formulated in continuous-time with the aid of binary variables. A time-discretization turns the optimization problem into a mixed integer linear program (MILP). Unlike the cell-based approaches (Daganzo, 1995; Lin and Wang, 2004; Lo, 1999b), the proposed framework does not require modeling or computation within a link, thus reducing the number of (binary) variables and computational effort. The proposed model is free of vehicle-holding problems, and captures important features of signalized networks such as physical queue, spill back, vehicle turning, time-varying flow patterns and dynamic signal timing plans. The MILP can be efficiently solved with standard optimization software.

This paper is concerned with a dynamic traffic network performance model, known as dynamic network loading (DNL), that is frequently employed in the modeling and computation of analytical dynamic user equilibrium (DUE). As a key component of continuoustime DUE models, DNL aims at describing and predicting the spatial-temporal evolution of traffic flows on a network that is consistent with established route and departure time choices of travelers, by introducing appropriate dynamics to flow propagation, flow conservation, and travel delays. The DNL procedure gives rise to the path delay operator, which associates a vector of path flows (path departure rates) with the corresponding path travel costs. In this paper, we establish strong continuity of the path delay operator for networks whose arc flows are described by the link delay model (Friesz et al., 1993). Unlike result established in Zhu and Marcotte (2000), our continuity proof is constructed without assuming a priori uniform boundedness of the path flows. Such a more general continuity result has a few important implications to the existence of simultaneous route-and-departure choice DUE without a priori boundedness of path flows, and to any numerical algorithm that allows convergence to be rigorously analyzed.

This paper is concerned with highway traffic estimation using traffic sensing data, in a Lagrangian-based modeling framework. We consider the Lighthill-Whitham-Richards (LWR) model (Lighthill and Whitham, 1955; Richards, 1956) in Lagrangian-coordinates, and provide rigorous mathematical results regarding the equivalence of viscosity solutions to the Hamilton-Jacobi equations in Eulerian and Lagrangian coordinates. We derive closed-form solutions to the Lagrangian-based Hamilton-Jacobi equation using the Lax-Hopf formula (Daganzo, 2005; Aubin et al., 2008), and discuss issues of fusing traffic data of various types into the Lagrangian-based H-J equation. A numerical study of the Mobile Century field experiment (Herrera et al., 2009) demonstrates the unique modeling features and insights provided by the Lagrangian-based approach.

Dynamic user equilibrium (DUE) is the most widely studied form of dynamic traffic assignment (DTA), in which road travelers engage in a non-cooperative Nash-like game with departure time and route choices. DUE models describe and predict the time-varying traffic flows on a network consistent with traffic flow theory and travel behavior. This paper documents theoretical and numerical advances in synthesizing traffic flow theory and DUE modeling, by presenting a holistic computational theory of DUE, which is numerically implemented in a MATLAB package. In particular, the dynamic network loading (DNL) sub-problem is formulated as a system of differential algebraic equations based on the Lighthill-Whitham-Richards fluid dynamic model, which captures the formation, propagation and dissipation of physical queues as well as vehicle spillback on networks. Then, the fixed-point algorithm is employed to solve the DUE problems with simultaneous route and departure time choices on several large-scale networks. We make openly available the MATLAB package, which can be used to solve DUE problems on user-defined networks, aiming to not only facilitate benchmarking a wide range of DUE algorithms and solutions, but also offer researchers a platform to further develop their own models and applications. The MATLAB package and computational examples are available at https:// github.com/DrKeHan/DTA.

To optimize revenue, service firms must integrate within their pricing policies the rational reaction of customers to their price schedules. In the airline or telecommunication industry, this process is all the more complex due to interactions resulting from the structure of the supply network. In this paper, we consider a streamlined version of this situation where a firm's decision variables involve both prices and investments. We model this situation as a joint design and pricing problem that we formulate as a mixed-integer bilevel program, and whose properties are investigated. In particular, we take advantage of a feature of the model that allows the development of an algorithmic framework based on Lagrangean relaxation. This approach is entirely novel, and numerical results show that it is capable of solving problems of significant sizes.

"Revenue optimization in energy networks involving self-scheduled demand and a smart grid" Forthcoming, Computers and Operations Research 134 (2021) 2. M. Zimmermann, E. Frejinger and P. Marcotte "A strategic Markovian traffic equilibrium model for capacitated networks" Transportation Science 5 (2021) 574-591 3. T. Dan, A. Lodi and P. Marcotte "An exact algorithmic framework for a class of mixed-integer programs with equilibrium constraints"

We study the Demand Adjustment Problem (DAP) associated to the urban traffic planning. The framework for the formulation of the DAP is mathematical programming with equilibrium constraints. In particular, if we consider the optimization problem equivalent to the equilibrium problem, the DAP becomes a bilevel optimization problem. In this work we present a descent scheme based on the approximation of the gradient of the objective function of DAP.

We consider a liberalized electricity market, divided in zones interconnected by capacitated transmission links, where a large dimensional power producer operates. We introduce a model for determining the optimal bidding strategies of the large dimensional producer, so as to maximize his own market share, while guaranteeing an annual profit target and satisfying technical constraints. The model determines the optimal medium-term resource scheduling and yields the hourly zonal electricity prices, as it includes constraints representing the Market Clearing process. In order to compute the global solution, the complementarity conditions are formulated as mixed integer linear constraints and the revenue terms are expressed by piece-wise linear functions. The model can be used for analyzing the behavior of market prices in electricity markets where a large dimensional producer can exert market power. It can also be used by investors as a simulation tool for evaluating both the impact on the market and the profitability of investment decisions in the zonal electricity market. A case study related to the Italian electricity market is discussed.

By using a hypothetical transport network that reflects common origin and destination relations in a regional transport network, we illustrate the effects of changing fares from a zonal to a distance-based structure. We take the zonal fare as a base case and model the effect of different fare/km, including non-additive fares, where the marginal price per km is decreasing with a typical regional network. We restrict our analysis to a fixed total demand and consider the effects of fares on route choice including station access choice and walking to nearby destinations. The results indicate some general trends that can be expected, such as the fare range in order to achieve similar fare revenue incomes. At this fare parity point the total travel time tends to be reduced in the distance-based case but the flows become less dispersed. Furthermore, in case of a non-additive distance-based fare, we show that total utility could be improved at the fare parity point compared to additive fares.

Non-additive link cost functions are common and important for a range of assignment problems. In particular in transit assignment, but also a range of other problems path splits further need to consider node cost uncertainties leading to the notion of hyperpaths. We discuss the problem of finding optimal hyperpaths under non-additive link cost conditions assuming a cost vector with a limited number of marginal decreasing costs depending on the number of links already traversed. We illustrate that these non-additive costs lead to violation of Bellman's optimality principle which in turn means that standard procedures to obtain optimal destination specific hyperpath trees are not feasible. To overcome the problem we introduce the concepts of a "travel history vector" and active critical, passive critical and fixed nodes. The former records the expected number of traversed links until a node, and the latter distinguishes nodes for which the cost can be determined deterministically. With this we develop a 2stage solution approach. In the first stage we test whether the optimal hyperpath can be obtained by backward search. If this is not the case, we propose a so called "selective hyperpath generation" among hyperpaths to a (small) number of active critical nodes and combine this with standard hyperpath search. We illustrate our approach by applying it to the Sioux Falls network showing that even for link cost functions with large step changes we are able to obtain optimal hyperpaths in a reasonable computational time.

Recently, simulation-based methods have been successfully used for solving challenging stochastic optimization problems and equilibrium models. Here we report some of the recent progress we had in broadening the applicability of so-called the sample-path method to include the solution of certain stochastic mathematical programs with equilibrium constraints. We first describe the method and the class of stochastic mathematical programs with complementarity constraints that we are interested in solving and then outline a set of sufficient conditions for its almostsure convergence. We also illustrate an application of the method to solving a toll pricing problem in transportation networks. These developments also make it possible to solve certain stochastic bilevel optimization problems and Stackelberg games, involving expectations or steady-state functions, using simulation.

In this note we give an elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints. The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of Farkas lemma and the Bolzano-Weierstrass property for compact sets.

We consider equilibrium constrained optimization problems, which have a general formulation that encompasses well-known models such as mathematical programs with equilibrium constraints, bilevel programs, and generalized semi-infinite programming problems. Based on the celebrated KKM lemma, we prove the existence of feasible points for the equilibrium constraints. Moreover, we analyze the topological and analytical structure of the feasible set. Alternative formulations of an equilibrium constrained optimization problem (ECOP) that are suitable for numerical purposes are also given. As an important first step for developing efficient algorithms, we provide a genericity analysis for the feasible set of a particular ECOP, for which all the functions are assumed to be linear. KEYWORDS. Equilibrium problems, existence of feasible points, mathematical programs with equilibrium constraints, problems with complementarity constraints, bilevel programs, generalized semiinfinite programming, genericity.

We address the problem of developing optimal bidding strategies for an energy producer that participates in a day-ahead electricity market. The producer is assumed to have complete knowledge of the market's parameters, i.e., the demand for energy and the bids/costs of the other producers. The problem is formulated as a mixed integer bilevel optimization model, with the individual producer maximizing his profit at the upper level objective function. At the lower level, an independent system operator clears the market and determines the energy dispatch of each participating producer that satisfies the demand for energy at the minimum total system bid-cost. Discrete variables that are used to represent the commitment of the production units prohibit the application of traditional methodologies for the solution of the problem, such as the substitution of the lower level problem by its first order optimality conditions. We present an exact algorithm based on mixed integer parametric programming for obtaining the optimal solution of the problem, and we illustrate its application on a small case study.

In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form 0 ∈ G(x, y) + Q(x, y), where both mappings G and Q are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal solutions requires quite restrictive assumptions. Our techniques are mainly based on modern tools of variational analysis and generalized differentiation revolving around the fundamental extremal principle in variational analysis and its analytic counterpart known as the subdifferential variational principle.

A new algorithm is presented for use in conjunction with the imposed rotation method as reformulated by Kaneko as a (n x 2n) linear complementarity problem. The inelastic trilinear moment rotation law, which is modelled to reflect behaviour at critical sections, is adjusted during the incremental analysis in order to adhere the non-holonomic behaviour of reinforced concrete (RC) when stress unloading occurs. The algorithm is utilized in performing inelastic analysis for the optimal design of RC frames. Optimum concrete dimensions and reinforcement are evaluated for minimum cost. Optimization, following the multilevel approach. is carried out on typical frames using SUMT and Rosenbrock minimization methods.

The planning of a rail transit system is a complex process involving the determination of station locations and the rail line alignments connecting the stations. There are many requirements and constraints to be considered in the planning process, with complex correlations and interactions, necessitating the application of optimization models in order to realize optimal (i.e. reliable and cost-effective) rail transit systems. Although various optimization models have been developed to address the rail transit system planning problem, they focus mainly on the planning of a single rail line and are therefore, not appropriate in the context of a multi-line rail network. In addition, these models largely neglect the complex interactions between station locations and associated rail lines by treating them in separate optimization processes. This thesis addresses these limitations in the current models by developing an optimal planning method for multiple lines, taking into account the relevant influencing factors, in a single integrated process using a geographic information system (GIS) and a genetic algorithm (GA). The new method considers local factors and the multiple planning requirements that arise from passengers, operators and the community, to simultaneously optimize the locations of stations and the associated line network linking them. The new method consists of three main levels of analysis and decision-making. Level I identifies the requirements that must be accounted for in rail transit system planning. This involves the consideration of the passenger level of service, operator productivity and potential benefits for the community. The analysis and decision making process at level II translates these requirements into effective criteria that can be used to evaluate and compare alternative solutions. Level III formulates mathematical functions for these criteria, and incorporates them into a single planning platform within the context of an integrated optimization model to achieve a rail transit system that best fits the desired requirements identified at level I. This is undertaken in two main stages. Firstly, the development of a GIS based algorithm to screen the study area for a set of feasible station locations. Secondly, the use of a heuristic optimization algorithm, based on GA to identify an optimum set of station locations from the pool of feasible stations, and, together with the GIS system, to generate the line network connecting these stations. The optimization algorithm resolves the essential trade-off between an effective rail system that provides high service quality and benefits for both the passenger and the whole community, and an economically efficient system with acceptable capital and operational costs. Imperial College London vi Chro Ahmed The proposed integrated optimization model is applied to a real world case study of the City of Leicester in the UK. The results show that it can generate optimal station locations and the related line network alignment that satisfy the various stakeholder requirements and constraints.

SOMMARIO : Si studiano atlraverso l'analisi limite le piastre coslituite di materiale rigido-plastico aventi superficie di snervamento di tipo poliedrico. I principi di eslremo riguardanti la vatutazione della potenza specifica di dissipazione vengono definiti attraverso i concelti della programmazione lineare. Altraverso il onto Ieorema cinematico dell'analisi limite, si formula la legge cbe regola il co/lasso plastico delle piastre. Viene proposlo un procedimento per la determinazione approssimata del carico di collasso del turin generale. Si conclude infine con una breve indagine numerica relativa al caso di piastra qnadrata con carico uniforme. SUMMARY: Rigid-plastic plates having a piecewise linear yield surface are studied O' limit attalysis. Extremum principles for the evaluation of specific dissipation power are defined by means of linear programming concepts. The law governing tbe plastic collapse of plates is formulated on the basis of the wellknown kinematic theorem of limit analysis. A general procedure for/be approximate determination of the collapse load is proposed. The paper ends lvitb a brief numeric investigation of the uniform O, loaded square plate.

This paper aims at studying a broad class of mathematical programming with non-differentiable vanishing constraints. First, we are interested in some various qualification conditions for the problem. Then, these constraint qualifications are applied to obtain, under different conditions, several stationary conditions of type Karush/Kuhn-Tucker.

In this paper, we study the difficult class of optimization problems called the mathematical programs with vanishing constraints or MPVC. Extensive research has been done for MPVC regarding stationary conditions and constraint qualifications using geometric approaches. We use the Fritz John approach for MPVC to derive the M-stationary conditions under weak constraint qualifications. An enhanced Fritz John type stationary condition is also derived for MPVC, which provides the notion of enhanced M-stationarity under a new and weaker constraint qualification: MPVC-generalized quasinormality. We show that this new constraint qualification is even weaker than MPVC-CPLD. A local error bound result is also established under MPVC-generalized quasinormality.

In this paper, we analyze infinite discrete-time games between hydraulic and thermal power operators in the wholesale electricity market. Two types of games are considered: Cournot closed-loop game and Stackelberg closed-loop game. W e consider a deregulated electrical industry where certain demand is satisfied by hydraulic and thermal technologies. The hydraulic operator decides the production in each season of each period that maximizes the sum of expected profit from power generation with respect to the stochastic dynamic constraint on the water stored in the dam, the environmental constraint and the non-negative output constraint. In contrast, the thermal plant is operated with quadratic cost function, with respect to the capacity production constraint and the non negativity output constraint. This paper is devoted to the numerical computations of equilibrium strategies and value function in each kind of games. We show that under imperfect competition, the hydraulic operator has a strategic storage of water in the peak season. Then, we quantify the strategic inter annual and intra annual water transfer in the both games and we compare the numerical results. Under Cournot closed-loop game, we show that the traditional principle of least-cost operation is inverted at the binding capacity constraint of thermal operator. Finally, under Stackelberg closed-loop game, we show that thermal operator can restrict the hydraulic output without compensation. The technical complementarities and Stackelberg competition may distorted the traditional "merit order" operating principal.

Demand response (DR) using shared energy storage systems (ESSs) is an appealing method to save electricity bills for users under demand charge and time-of-use (TOU) price. A novel Stackelberg-game-based ESS sharing scheme is proposed and analyzed in this study. In this scheme, the interactions between selfish users and an operator are characterized as a Stackelberg game. Operator holds a large-scale ESS that is shared among users in the form of energy transactions. It sells energy to users and sets the selling price first. It maximizes its profit through optimal pricing and ESS dispatching. Users purchase some energy from operator for the reduction of their demand charges after operator’s selling price is announced. This game-theoretic ESS sharing scheme is characterized and analyzed by formulating and solving a bi-level optimization model. The upper-level optimization maximizes operator’s profit and the lower-level optimization minimizes users’ costs. The bi-level model is transformed and linearized into a mixed-integer linear programming (MILP) model using the mathematical programming with equilibrium constraints (MPEC) method and model linearizing techniques. Case studies with actual data are carried out to explore the economic performances of the proposed ESS sharing scheme.

This document describes the models for network design and planning in the field of urban transportation that have been developed in the course of research project TRA2008-06782-C02-02. This includes transit, rapid transit and bus networks. This project can be considered in many aspects as a follow up of previous research project TRA2005-09068-C03-02/MODAL. The network topology design is being considered in this project as well as logistic problems regarding vehicle assignment and timetabling. Also, the integration of different stages traditionally carried out in classical approaches of the modelization process (i.e. design and planning) is done via unified and compact models contributing in this way to the improvement in the consistency of the solutions. Some mathematical programming models are exposed in order to set the number of services at previously specified bus lines, which are intended to assist high demand occurring during the disruption of the Rapid Transit services or during the celebration of massive events. By means of this model two types of basic magnitudes can be determined, basically: a) the number of bus units assigned to each line and b) the number of services that should be assigned to those units. The model can be considered of the system optimum type, in the sense that the assignment of units and of services is carried out minimizing a linear combination of operation costs and total travel time of users. The model considers delays experienced by buses as a consequence of the get in/out of the passengers, queueing at stations and the delays that passengers experience waiting at the stations.