Bi-level Mathematical Programming Research Articles

Flat-jack tests and inverse analysis for the identification of stress states and elastic properties in concrete dams

In this paper an experimental-numerical method, centred on flat-jack tests, is presented for the identification of local stress states and possibly deteriorated elastic properties of concrete in existing dams. It is shown herein that the synergistic combination of new pattern of flat-jack experiment, computer simulation of the test (by conventional finite elements) and inverse analysis allows to exploit experimental data more effectively than by the traditional procedure, and to achieve more information on material properties. In fact, at suitably chosen locations on the free surface of the monitored dam, all the components of local (plane) stress state and the elastic moduli in two orthogonal directions (including shear stiffness) can be estimated by the proposed method. The inverse problem in point is formulated as a sequence of two parameter identifications, i.e. as a bilevel (in the sense of Stackelberg) mathematical programming problem. The solution in a stochastic context is achieved by means of a modified Bayes technique, allowing to obtain, in a “batch” (non sequential) way, parameter estimates endowed with a covariance matrix which quantifies their degrees of confidence and correlations.

The Formulation of the Optimal Strategies for the Electricity Producers Based on the Particle Swarm Optimization Algorithm

In competitive electricity markets, the producer as a market participant strives to find the optimal supply function with the objective of maximizing his/her producer surplus in the market clearing. The model of the producer surplus maximization is a bilevel mathematical programming problem within which the market clearing is taken into account. By using the deterministic approaches, it is difficult to obtain the global solution of the bilevel optimization problem, even for a single hourly market clearing. This is due to the fact that the objective function of such a problem is not concave, and there are nonlinear complementarity terms introduced by using the KKT conditions to represent the market clearing. When the bilevel optimization problem is modified to consider multiple hourly market clearings, such as to maximize the total producer surplus in one day, solving such a problem is almost intractable. A heuristic approach should be another option. For its simplicity and immunity to the local optimum, the particle swarm optimization (PSO) algorithm is employed in this paper to find the optimal supply function of the electricity producer. Based on the IEEE 30-bus test system, different simulation cases with respect to a single hourly market clearing and a daily market clearing are tested to show the efficiency and robustness of the PSO algorithm. In addition, the parameterization techniques used in formulating the optimal supply function are analyzed based on the simulation results

A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problem

Solving optimization problems with multiple objectives under uncertainty is generally a very difficult task. Evolutionary algorithms, particularly genetic algorithms, have shown to be effective in solving this type of complex problems. In this paper, we develop a simulation-based multi-objective genetic algorithm (SMOGA) procedure to solve the build-operate-transfer (BOT) network design problem with multiple objectives under demand uncertainty. The SMOGA procedure integrates stochastic simulation, a traffic assignment algorithm, a distance-based method, and a genetic algorithm (GA) to solve a multi-objective BOT network design problem formulated as a stochastic bi-level mathematical program. To demonstrate the feasibility of SMOGA procedure, we solve two mean-variance models for determining the optimal toll and capacity in a BOT roadway project subject to demand uncertainty. Using the inter-city expressway in the Pearl River Delta Region of South China as a case study, numerical results show that the SMOGA procedure is robust in generating ‘good’ non-dominated solutions with respect to a number of parameters used in the GA, and performs better than the weighted-sum method in terms of the quality of non-dominated solutions.

Bi-Level Mathematical Programming Model for Locating Bus Stops and Optimizing Frequencies

This paper proposes a model for locating bus stops and optimizing bus frequencies in congested local public transport networks. The analysis addresses the issue of reducing the overall costs of operating a transport system, including costs associated with provision of services and bus stop construction, using a bi-level mathematical programming method. The upper level defines the overall cost of the system, which must be kept to a minimum taking into account operational restrictions, and the lower level defines a behavioral model for system users. The suggested model mixes optimization and simulation and allows supply and demand to be linked; this ensures consistency between equilibrium flows and bus frequencies and between equilibrium flows and the distances between bus stops calculated at each iteration of the algorithm.

Bi-Level Mathematical Programming Model for Locating Bus Stops and Optimizing Frequencies

This paper proposes a model for locating bus stops and optimizing bus frequencies in congested local public transport networks. The analysis addresses the issue of reducing the overall costs of operating a transport system, including costs associated with provision of services and bus stop construction, using a bi-level mathematical programming method. The upper level defines the overall cost of the system, which must be kept to a minimum taking into account operational restrictions, and the lower level defines a behavioral model for system users. The suggested model mixes optimization and simulation and allows supply and demand to be linked; this ensures consistency between equilibrium flows and bus frequencies and between equilibrium flows and the distances between bus stops calculated at each iteration of the algorithm.

A bi-level model of the relationship between transport and residential location

This paper explores a bid-rent network equilibrium model which represents the relationship between transport and residential location. The relationship is examined in terms of the competition of decision-makers for locations. The model discusses difficulties in addressing the characteristics of locations, particularly heterogeneity and indivisibility. A hedonic interpretation is included as a way to resolve these challenges. The model investigates the process in which households make their decisions. This process is shown as an n-player non-cooperative game, following the Nash equilibrium for this game, which is defined as well. The game is accompanied by the systematic interactions between transport and land-use. A mutual adjustment process represents these interactions. The three components are structured by a bi-level mathematical program. The final formulation is interpreted as an oligopolistic Cournot game of which consequence is an approximation of the n-player non-cooperative game. The functional relationship between the decision variables of the upper and the lower levels of the bi-level program produces endogenously-determined transport impedance and locational attractiveness. The endogenous network performance indices of the model are expected to overcome the lack of a realistic network equilibrium description in the existing models. A path-based heuristic algorithm and a simple numerical example are presented. Finally, some concluding remarks are given.

A Trust-Region Method for Nonlinear Bilevel Programming: Algorithm and Computational Experience

We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upper-level objective and all constraint-defining functions, as well as a quadratic approximation of the lower-level objective. We describe the main features of the algorithm and the resulting software. Numerical experiments tend to confirm the promising behavior of the method.

Mathematical models for applying cell suppression methodology in statistical data protection

This paper concerns the problem of protecting sensitive information in tabular data against different intruders. Statistical offices allow different schemes to solve this problem. One of them is the so-called cell suppression methodology, where some cell values can be suppressed. We present four mathematical models for the problem of finding a cell suppression pattern minimizing the loss of information while guaranteeing protection level requirements for different sensitive cells and different intruders. This problem is more general than, for example, the so-called “common respondent problem” mentioned by Jewett [Disclosure analysis for the1992 economic census; Working paper, United States Bureau of the Census, 1993] in statistical disclosure control. The first model corresponds to bi-level mathematical programming. The second model belongs to integer linear programming (ILP) and could be used on small-size tables where some nominal values are known to assume discrete values. The third model is also an ILP model which is valid when the nominal values of the table are continuous numbers, with the advantage of containing a small number of variables (one 0–1 variable for each cell in the table). On the other hand, this model has a bigger number of linear inequalities (related to the number of sensitive cells and the number of attackers). Nevertheless, this paper addresses this disadvantage which, when necessary, is overcome by a dynamic generation of the important inequalities. The overall algorithm follows a modern mathematical programming technique known as the branch-and-cut approach, and allows the finding of optimal solutions for medium-size tables. On large-size tables the approach can be used to find near-optimal solutions. The fourth model adds two continuous variables for each cell to the third model to allow the statistical office more control over the loss of information, thus producing more accurate but protected patterns. The paper ends pointing out another alternative methodology to produce patterns by shrinking all the different intruders into a single one, and compares it with the classical single-attacker methodology and with the above multi-attacker methodology.

Conceptual design of metallurgical processes based on thermodynamic and economic insights

This work illuminates a practical design approach for developing economically viable and thermodynamically feasible metallurgical manufacturing processes. Successively refined economic potential estimators serve as the driving force for systematically advancing the design effort. The focus on process economics aims at identifying design inferior options and discarding process configurations with marginal economic performance early in the development cycle. In the proposed approach, novel process ideas evolve gradually as prescribed by the decision hierarchy by Douglas [AIChE J. 31 (1985) 353] modified to address metallurgical processes. Due to lack of heuristics in modeling reaction networks, a bi-level mathematical program for the simultaneous optimization of economic performance and complex multi-phase thermodynamic equilibrium is developed. The discussion includes the derivation of a thermodynamically consistent framework for integrating empirical atom interaction coefficients into the total Gibbs free energy model. The article demonstrates the beneficial use of mathematical modeling and non-linear mathematical programming within a systematic design framework.

Application of lmmune Algorithms to Combinatorial Optimization for Traffic Safety Countermeasures

The planning of traffic safety countermeasures in urban road network is considered as a combinational opturnzation problem for the means and the spots of safety countermeasures to be introduced. It can be formulated as a bi-level mathematical program with traffic network equilibrium constraint. Application of immune algorithm (IA) to solving this optimization problem approximately is examined in this research. The optimization process of immune algorithm is discussed in comparison with genetic algorithm (GA). In this way, it is shown that immune algorithm is an effective algorithm to solve the combinational optimization problems in planning of traffic safety countermeasures.

Mean-Variance Model for the Build-Operate-Transfer Scheme Under Demand Uncertainty

A mean-variance model was developed for determining the optimal toll and capacity in a build-operate-transfer (BOT) roadway project subject to traffic demand uncertainty. This mean-variance model involves two objectives: maximizing mean profit and minimizing the variance (or standard deviation) of profit. The variance associated with profit is considered as a risk. Because maximizing expected profit and minimizing risk are often conflicting, there may not be a single best solution that can simultaneously optimize both objectives. Hence, it is necessary to explicitly consider this as a multiobjective problem so that a set of nondominated solutions can be generated. In this study, the optimal toll and capacity selection for the BOT problem under demand uncertainty is formulated as a special case of the stochastic network design problem. A simulation-based multiobjective genetic algorithm was developed to solve this stochastic bilevel mathematical programming formulation. Numerical results are also presented as a case study.

TRANSYT derivatives for area traffic control optimisation with network equilibrium flows

A bi-level mathematical programming technique has previously been proposed, in which an area traffic control problem is dealt with as the upper level problem whilst the users’ equilibrium traffic assignment is dealt with as the lower level problem. The performance index of the upper level problem is defined as the sum of a weighted linear combination of the rate of delay and number of stops per unit time for all traffic streams, which is evaluated by the traffic model from TRANSYT. Approximate mathematical expressions for various components of the performance index and the average delay to a vehicle at the downstream junction in the TRANSYT model for both undersaturated and oversaturated links are considered. As a step towards heuristics for finding good solutions to this form of the bi-level problem, in this paper, explicit expressions have been obtained for the partial derivatives of all these quantities with respect to the signal control variables. Numerical calculations have been shown on example test network and good agreement of the results with these derivatives determined by numerical differentiation is obtained.

Metallurgical Process DesignA Tribute to Douglas' Conceptual Design Approach

This case study illuminates a practical design problem for cleaner metallurgical manufacturing. The task consists of assessing the feasibility of a novel low-waste process capable of eliminating unwanted, hazardous waste streams in stainless steel production. The novel flowsheet evolves gradually as prescribed by the decision hierarchy of Douglas (Douglas, J. M. AIChE J. 1985, 31, 353−362). Successively refined economic potential estimators discriminate inferior options and discard process configurations with marginal economic performance early in the design cycle. The complex liquid metal reaction network is optimized via a multiphase total Gibbs free energy model. A bilevel mathematical program for the simultaneous optimization of economic performance at equilibrium within specified operational bounds is presented. The article uses the case study to elucidate methodological aspects in systematic decision making for metallurgical process synthesis. Examples for the beneficial use of mathematical modeling...

A Bilevel Model for Toll Optimization on a Multicommodity Transportation Network

We consider the problem of determining a set of optimal tolls on the arcs of a multicommodity transportation network. The problem is formulated as a bilevel mathematical program where the upper level consists in a firm that raises revenues from tolls set on arcs of the network, while the lower level is represented by a group of users travelling on shortest paths with respect to a generalized travel cost.

Contact shape optimization: a bilevel programming approach

We consider the problem of shape optimization of nonlinear elastic solids in contact. The equilibrium of the solid is defined by a constrained minimization problem, where the body energy functional is the objective and the constraints impose the nonpenetration condition. Then the optimization problem can be formulated in terms of a bilevel mathematical program. We describe new optimality conditions for bilevel programming and construct an algorithm to solve these conditions based on Herskovits' feasible direction interior point method. With this approach we simultaneously carry out shape optimization and nonlinear contact analysis. That is, the present method is a "one shot" technique. We describe some numerical examples solved in a very efficient way.

Two-Stage Approach for Quantitative Policy Analysis Using Bilevel Programming

Quantitative policy analysis problems with hierarchical decision-making can be modeled as bilevel mathematical programming problems. In general, the solution of these models is very difficult; however, special cases exist in which an optimal solution can be obtained by ordinary mathematical programming techniques. In this paper, a two-stage approach for the formulation, construction, solution, and usage of bilevel policy problem is presented. An outline of an example for analyzing Israel's public expenditure policy is also given.

Dependence of bilevel mathematical programs on irrelevant constraints

For standard, single-level mathematical programs, any optimal solution remains optimal when an inactive (i.e., irrelevant) constraint is added to the problem, a property termed independence of irrelevant constraints (IIC). In general, bilevel mathematical programs do not possess the IIC property. This paper presents necessary and sufficient conditions for the IIC property to hold for a bilevel program. A straightforward test is described that concludes whether or not a bilevel program is IIC.

Exact Penalization and Necessary Optimality Conditions for Generalized Bilevel Programming Problems

The generalized bilevel programming problem (GBLP) is a bilevel mathematical program where the lower level is a variational inequality. In this paper we prove that if the objective function of a GBLP is uniformly Lipschitz continuous in the lower level decision variable with respect to the upper level decision variable, then using certain uniform parametric error bounds as penalty functions gives single level problems equivalent to the GBLP. Several local and global uniform parametric error bounds are presented, and assumptions guaranteeing that they apply are discussed. We then derive Kuhn--Tucker-type necessary optimality conditions by using exact penalty formulations and nonsmooth analysis.

Incentive strategies with threats in dynamic constrained-stackelberg problems. a bilevel programming approach

Dynamic constrained-Stackelberg equilibrium problems are two player differential games, with constraints on decision variables, where the two players, namely the Leader and the Follower, are not in a symmetric position, becasue of their different access to information,and announce their strategies sequentially. Inparticular, the Leader dominates the entire game by imposing a solution which is favourable to himself using incentive strategies with threats In this paper the linear-quadratic case is considered, and a new resolution approach, based on methodologies pertaining to bilevel mathematical programming (BLP), is presented. The basic framework of the proposed approach has been validated by solving a wide range of randomly generated instances whose computational results are reported and discussed here

Algorithms for nonlinear bilevel mathematical programs

The bilevel programming problem (BLPP) is a model of a leader-follower game in which play is sequential and cooperation is not permitted. Some basic properties of the general model are developed, and a conjecture relevant to solution procedures is presented. Two algorithms are presented for solving various versions of the game when certain convexity conditions hold. One algorithm relies upon a hybrid branch-and-bound scheme and does not guarantee global optimality. Another is based on objective function cuts and, barring numerical stability problems with the optimizer, is guaranteed to converge to an epsilon -optimal solution. The performance of the two algorithms is examined using randomly generated test problems. The computational performance of the branch-and-bound algorithm is explored, and the cutting-plane algorithm is used to determine whether or not the branch-and-bound algorithm is uncovering global optima.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>