Large-scale Mathematical Programming Research Articles

Distortions of National Policies to Renewable Energy Cooperation Mechanisms

The EU endeavors to stimulate the use of renewable energy cooperation mechanisms. These cooperation mechanisms can significantly reduce the policy cost for meeting renewable targets. Several authors, however, have raised concerns that such cooperation mechanisms can be subject to efficiency losses due to different national regulatory conditions, and due to an ill-advised selection of cross-border support instruments. A quantitative evaluation of these effects remains missing. To address this gap, we first introduce a unifying analytical framework to show how optimal cross-border renewable energy trade should be organized and how these mechanisms could be distorted. We then develop a partial equilibrium model, formulated as a large-scale mathematical program with equilibrium constraints, to assess the impact of (i) different national grid cost allocation regimes and (ii) different cross-border feed-in premium implementations. Our results indicate that EU-wide auctions for renewable electricity should (i) not be based on sliding feed-in premiums and should (ii) ideally be discriminatory if national regulatory conditions differ across Member States. We also consider country-level distributional effects and confirm that Member States can lose when engaging in renewable cooperation.

An integer programming model and directed Steiner-forest based heuristic for routing less-than-truckload freight

Less-than-truckload (LTL) freight transportation is a vital part of Canada’s economy, with revenues running into billions of dollars and a cascading impact on many other industries. LTL operators have to deal with large volumes of shipments and uncertainty in demand patterns. In an industry that already has low profit margins, it is therefore vitally important to make good quality routing decisions without expending a lot of time. The optimization of such LTL freight networks results in large scale mathematical programming problems. In this paper, we present a novel Integer Linear Programming (ILP) formulation and heuristics for routing LTL freight. Experiments in collaboration with our industry partner indicate that our proposals can significantly lower routing costs for Canadian LTL operators. In addition, our approach can be useful for generating load plans on larger American LTL networks.

A friendly computable characteristic function

We consider an n-player game in coalitional form. We use the so-called δ characteristic function to determine the strength of all possible coalitions. The value of a coalition is obtained under the behavioral assumption that left-out players do not react strategically to the formation of that coalition, but stick to their Nash equilibrium actions in the n-player noncooperative game. This assumption has huge computational merit, especially in games where each player is described by a large-scale mathematical program. For the class of games with multilateral externalities discussed in Chander and Tulkens, we show that the δ characteristic function is superadditive and has a nonempty core, and that the δ-core is a subset of the γ-core.

Strict linear prices in non-convex European day-ahead electricity markets

The European power grid can be divided into several market areas where the price of electricity is determined in a day-ahead auction. Market participants can provide continuous hourly bid curves and combinatorial bids with associated quantities given the prices. The goal of our auction is to maximize the economic surplus of all participants subject to quantity constraints and price constraints. The price constraints ensure that no one incurs a loss. Only traders who submitted a combinatorial bid might miss a not-realized profit. The resulting problem is a large-scale mathematical program with equilibrium constraints (MPEC) and binary variables that cannot be solved efficiently by standard solvers. We present an exact algorithm and a fast heuristic for this type of problem. Both algorithms decompose the MPEC into a master problem (a mixed-integer quadratic program) and pricing subproblems (linear programs). The modelling technique and the algorithms are applicable to a wide variety of combinatorial auctions that are based on mixed-integer programs.

Contributors

Contributors

An extended mathematical programming framework

Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to facilitate higher level structure identification. The structures, which often involve constraints on the solution sets of other models or complementarity relationships, can be exploited by modern large scale mathematical programming algorithms for efficient solution. A specific implementation of this framework is outlined that communicates structure from the GAMS modeling system to appropriate solvers in a computationally beneficial manner. Example applications are taken from chemical engineering.

Leader-Follower Equilibria for Electric Power and NO x Allowances Markets

This paper investigates the ability of the largest producer in an electricity market to manipulate both the electricity and emission allowances markets to its advantage. A Stackelberg game to analyze this situation is constructed in which the largest firm plays the role of the leader, while the medium-sized firms are treated as Cournot followers with price-taking fringes that behave competitively in both markets. Since there is no explicit representation of the best-reply function for each follower, this Stackelberg game is formulated as a large-scale mathematical program with equilibrium constraints. The best-reply functions are implicitly represented by a set of nonlinear complementarity conditions. Analysis of the computed solution for the Pennsylvania–New Jersey–Maryland electricity market shows that the leader can gain substantial profits by withholding allowances and driving up NO x allowance costs for rival producers. The allowances price is higher than the corresponding price in the Nash–Cournot case, although the electricity prices are essentially the same.

An improved stepsize of the subgradient algorithm for solving the lagrangian relaxation problem

The Lagrangian relaxation approach has been successfully applied to many large-scale mathematical programming problems. The Lagrangian relaxation problem itself is a non-differentiable optimization problem. One of the methods for solving such problem is the subgradient algorithm. In this paper, we propose an improved stepsize of the subgradient algorithm for solving the Lagrangian relaxation problem. Our version of the algorithm may significantly improve the rate of convergence of subgradient algorithm when applied to the solution of Lagrangian relaxation problem. An illustrative numerical example is also given.

Column Generation Algorithms for Nonlinear Optimization, I: Convergence Analysis

Column generation is an increasingly popular basic tool for the solution of large-scale mathematical programming problems. As problems being solved grow bigger, column generation may however become less efficient in its present form, where columns typically are not optimizing, and finding an optimal solution instead entails finding an optimal convex combination of a huge number of them. We present a class of column generation algorithms in which the columns defining the restricted master problem may be chosen to be optimizing in the limit, thereby reducing the total number of columns needed. This first article is devoted to the convergence properties of the algorithm class, and includes global (asymptotic) convergence results for differentiable minimization, finite convergence results with respect to the optimal face and the optimal solution, and extensions of these results to variational inequality problems. An illustration of its possibilities is made on a nonlinear network flow model, contrasting its convergence characteristics to that of the restricted simplicial decomposition (RSD) algorithm.

Analysis of Hybrid Heuristic Model Reduction Algorithms for Solving Linear Programming Problems

Structural redundancies in large scale mathematical programming models are nothing uncommon and large scale Linear Programming Problems (LPP) are no exception. The presence of such embedded redundancies is hardly disputed and can play havoc with LP solution procedures and greatly hamper the solution effort. It is necessary to understand the modelling implications of these embedded redundancies as well as to exploit them during actual computation. The latter goal places heavy emphasis on efficient as well as effective identification techniques for economic application to large scale models. These embedded redundancies may be due to poor modelling and occasionally by bad source data.

From data to model and back to data: A bond portfolio management problem

The bond portfolio management problem is formulated as a multiperiod stochastic program using interest rate scenarios. The scenarios are sampled from the binomial lattice from a Black–Derman–Toy model. The paper analyzes the sensitivity of the solution of the resulting large-scale mathematical program with respect to the model inputs. The numerical results are for the Italian bond market.

Stability Properties of a Bond Portfolio Management Problem

The bond portfolio management problem is formulated as a multiperiod two-stage or multistage stochastic program based on interest rate scenarios. These scenarios depend on the available market data, on the applied estimation and sampling techniques, etc., and are used to evaluate coefficients of the resulting large scale mathematical program. The aim of the contribution is to analyze stability and sensitivity of this program on small changes of the coefficients – the (scenario dependent) values of future interest rates and prices. We shall prove that under sensible assumptions, the scenario subproblems are stable linear programs and that also the optimal first-stage decisions and the optimal value of the considered stochastic program possess acceptable continuity properties.

Convergence aspects of adaptive clustering in variable aggregation

A convergent iterative aggregation procedure is described. The procedure makes use of both weight updating and reclustering of variables during the iterative process. Scope and purpose During the last two decades, aggregation/disaggregation has become an important tool in operations research, both for modeling purposes and for providing approximate solutions of large-scale mathematical programs. Methods for iteratively improving an aggregate model (iterative aggregation) have recently got a lot of interest. The present paper is a contribution in that context. A convergent iterative aggregation procedure is developed, which in principle can be used not only to find an approximate solution of a large-scale linear program, but also to find the exact solution.

Optimal experimental methods of the large-scale mathematical programming and applications

The optimal experimental methods of a multidimensional dynamic programming, a large-scale linear and a complex nonlinear system are presented, which achieve a great step forward in the field of systems science. And excellent results are obtained in the applications of complex water resource systems.

Lower bounds for nonlinear assignment problems using many body interactions

This paper concerns lower bounding techniques for the general α-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space. This process involves two steps — (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated β times on an α-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent (α + β)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space. It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving O( N 2( α+ β−1) ) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP's. We illustrate all these concepts on the quadratic assignment problems With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP's of size 32 and have also improved upon existing lower bounds for other QAP's.

Database structures for mathematical programming models

In the design and use of large-scale mathematical programming systems, a substantial portion of the effort has no direct relation to the variables and constraints, but is instead concerned with the description, manipulation and display of data. Established principles of database design do not apply directly to mathematical programming, however, because there are significant differences of organization and content between the data for an optimization model and the data for a conventional database application such as payroll or order entry. The goal of this work is thus to derive and elucidate fundamental principles of database construction for the specific case of large-scale mathematical programming. Alternative formulations of a steel mill planning model, combining aspects of production and network linear programming, are presented as an example; these formulations are shown to correspond to relational and hierarchical database schemes that have contrasting strengths and weaknesses. A particular implementation of a database system for steel optimization is then introduced and discussed, and a variety of promising generalizations are surveyed.

The implementation of linear programming algorithms based on homotopies

A fundamental homotopy-based linear programming algorithm, which utilizes Euler-predictor and Newton-corrector steps with restarts, is formulated and investigated numerically on problems representative of linear programs that arise in practice. A rich array of refinements of this basic algorithm are possible within the homotopy framework. Such refinements are needed in any practical implementation and are discussed in detail. Implications for the design of integrated large-scale mathematical programming software are also briefly considered.

Particular solutions of the PDF-to-ODF inversion problem of texture analysis by large-scale mathematical programming

Discretizing the mathematically ill-posed inversion problem of texture analysis to calculate an orientation density defined on G ⊂ SO(3) from given pole densities defined on S3+ ⊂ \bb R3 results in a largely column rank deficient system of linear equations. Particular model solutions of this system which minimize or maximize, respectively, given objective functions like texture index or entropy are calculated and compared for the Sante Fe texture problem. It is shown that maximization of the texture index is generally not a device for conditional ghost correction. Furthermore, particular solutions with extreme values for a given texture component g0 ∈ Gn0 ∈ (Gn)n = 1,⋯N of the partition of the (Euler) orientation space G are calculated and discussed. Numerical evidence is given for the conjecture that application of a positive lower bound 0 < b to all orientation density functions (ODFs) f ∈ \scr F that are feasible with respect to a given set \scr P of pole density functions (PDFs), i.e. restriction to feasible ODFs 0 < b ≤ f(g), g ∈ G, f ∈ \scr F with a given minimum orientation density or uniform portion, respectively, substantially diminishes their variation. Finally, it is shown that this approach of optimizing individual texture components generalizes to the numerical determination of the variation width of feasible ODFs with respect to a given set \scr P of PDFs and the discretization of the inversion problem.

Fuzzy programming for large-scale multiobjective linear programming problems with block angular structure

Since Dantzig and Wolfe proposed the decomposition principle for the large-scale linear programming problem with the block angular structure in the early 1960's, there have been made a large number of intensive studies for the large-scale mathematical programming problems with special structures. Considering the fuzziness of the human decision, however, it seems more adequate to assume that the decisionmaker has the fuzzy goals or fuzzy constraints for the objective functions and for the coupled constraints in the problem formulated as the large-scale linear programming problem with the block angular structure. From such a viewpoint, this paper focuses on the large-scale multiobjective linear programming problem and proposes a new method to derive easily the satisficing solution for the decisionmaker considering his/her fuzzy goal. In the proposed method, the fuzzy goal of the decisionmaker for each objective function is specified by the linear membership function through the interaction with the decisionmaker. Fuzzy decision is applied to integrate the fuzzy goals. Then the original problem is decomposed into a master problem and several independent small-scale linear programming subproblems. The Dantzig-Wolfe decomposition principle is applied, where the optimal solution as a whole is obtained through iterations, by updating the goal function of each subproblem according to the solution of the master problem. Thus, it is shown that the satisficing solution is obtained relatively easily considering the fuzzy goals of the decisionmaker.

Production planning via scenario modelling

Several Linear Programming (LP) and Mixed Integer Programming (MIP) models for the production and capacity planning problems with uncertainty in demand are proposed. In contrast to traditional mathematical programming approaches, we use scenarios to characterize the uncertainty in demand. Solutions are obtained for each scenario and then these individual scenario solutions are aggregated to yield a nonanticipative or implementable policy. Such an approach makes it possible to model nonstationarity in demand as well as a variety of recourse decision types. Two scenario-based models for formalizing implementable policies are presented. The first model is a LP model for multi-product, multi-period, single-level production planning to determine the production volume and product inventory for each period, such that the expected cost of holding inventory and lost demand is minimized. The second model is a MIP model for multi-product, multi-period, single-level production planning to help in sourcing decisions for raw materials supply. Although these formulations lead to very large scale mathematical programming problems, our computational experience with LP models for real-life instances is very encouraging.