Pooling Problem Research Articles

APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes

Abstract Our recent work globally optimized two classes of large-scale pooling problems: a generalized pooling problem treating the network topology as a decision variable and an extended pooling problem incorporating environmental regulations into constraints. The pooling problems were optimized using a piecewise linear scheme that activates appropriate under- and overestimators with a number of binary decision variables that scales linearly with the number of segments in the piecewise relaxation. Inspired by recent work (Vielma & Nemhauser, 2010; Vielma, Ahmed, & Nemhauser, 2010b), we introduce a formulation for the piecewise linear relaxation of bilinear functions with a logarithmic number of binary variables and computationally compare the performance of this new formulation to the best-performing piecewise relaxations with a linear number of binary variables. We have unified our work by developing APOGEE, a computational tool that globally optimizes standard, generalized, and extended pooling problems. APOGEE is freely available to the scientific community at helios.princeton.edu/APOGEE/.

Distributive Politics and Electoral Incentives: Evidence from Seven US State Legislatures

We study the effect of electoral incentives on the allocation of public services across legislative districts. We develop a model in which elections encourage individual legislators to cater to parochial interests and thus aggravate the common pool problem. Using unique data from seven US states, we study how the amount of funding that a legislator channels to his district changes when he faces a term limit. We find that legislators bring less state funds to their district when they cannot run for re-election. Consistent with the Law of 1/N, this tendency is less pronounced in states with many legislative districts.

Stochastic pooling problem for natural gas production network design and operation under uncertainty

AbstractProduct quality and uncertainty are two important issues in the design and operation of natural gas production networks. This paper presents a stochastic pooling problem optimization formulation to address these two issues, where the qualities of the flows in the system are described with a pooling model and the uncertainty in the system is handled with a multi‐scenario, two‐stage stochastic recourse approach. In addition, multi‐objective problems are handled via a hierarchical optimization approach. The advantages of the proposed formulation are demonstrated with case studies involving an example system based on Haverly's pooling problem and a real industrial system. The stochastic pooling problem is a potentially large‐scale nonconvex Mixed‐Integer Nonlinear Program (MINLP), and a rigorous decomposition method developed recently is used to solve this problem. A computational study demonstrates the advantage of the decomposition method over a state‐of‐the‐art branch‐and‐reduce global optimizer, BARON. © 2010 American Institute of Chemical Engineers AIChE J, 2011

An enhanced energy conserving time stepping algorithm for frictionless particle contacts

AbstractAn Enhanced Energy Conserving Algorithm (EECA) formulation for time integration of frictionless contact–impact problems is presented. In it the energy, linear and angular momentum are conserved for every contact using an enhanced Penalty method. Previous formulations for these problems have shown that the total bodies' energy decreases for contact due to an artificial energy transfer between the penalty spring and the contacting bodies. Consequently, they are not able to reproduce a physical response after a single contact, introducing errors in trajectories and velocities.Through the conserving balance equations, EECA computes a physical response by inserting for every contact an additional amount of linear momentum and contact force. The structure of these equations defines the additional linear momentum to restore the energy and the enhanced Penalty method based on a spring and a dashpot. This method approximately enforces the first and second Kuhn–Tucker conditions.The new algorithm has been applied to several frictionless rigid problems using the Discrete Element Method. The first two problems consist of the simulation and analytical comparison of the Newton's Cradle and Carom problems (billiard pool problem). The last two are the hopper filling process and the breaking of a pool ball's triangular arrangement, both of which involve a medium number of contacts. Application of this formulation will be straightforward to elastic and general‐shaped bodies using the Finite Element Method. Copyright © 2010 John Wiley & Sons, Ltd.

Global Optimization of Large-Scale Generalized Pooling Problems: Quadratically Constrained MINLP Models

The generalized pooling problem is a generic way of identifying a topology containing sources nodes, intermediate storage tanks, and sinks when monitoring specific stream qualities is imperative. One important application domain of the generalized pooling problem is wastewater treatment. Choosing among the wide array of available wastewater treatment technologies is a combinatorially complex optimization problem that requires nonconvex terms to monitor regulated stream qualities about a treatment plant. In this work, we address five instantiations of the generalized pooling problem to global optimality by introducing (i) a quadratically constrained MINLP model formulation that reduces the number of bilinear terms, (ii) novel piecewise underestimation methods for the nonconvex bilinear terms to tighten the relaxation [Gounaris et al. Ind. Eng. Chem. Res. 2009, 48, 5742−5766; Wicaksono and Karimi AIChE J. 2008, 54, 991−1008], and (iii) a branch-and-bound algorithm suited to address the combinatorial complex...

Censored exploration and the dark pool problem

Dark pools are a recent type of stock exchange in which information about outstanding orders is deliberately hidden in order to minimize the market impact of large-volume trades. The success and proliferation of dark pools have created challenging and interesting problems in algorithmic trading---in particular, the problem of optimizing the allocation of a large trade over multiple competing dark pools. In this work, we formalize this optimization as a problem of multi-venue exploration from censored data, and provide a provably efficient and near-optimal algorithm for its solution. Our algorithm and its analysis have much in common with well-studied algorithms for managing the exploration--exploitation trade-off in reinforcement learning. We also provide an extensive experimental evaluation of our algorithm using dark pool execution data from a large brokerage.

Mathematical modeling and global optimization of large-scale extended pooling problems with the (EPA) complex emissions constraints

Environmental Protection Agency (EPA) Title 40 Code of Federal Regulations Part 80.45: Complex Emissions Model [40CFR80.45, 2007] codifies a mathematical model of gasoline emissions for reformulated gasoline (RFG) as a function of eleven fuel properties. In this paper we propose an extended pooling problem to maximize the profit of blending reformulated gasoline on a predetermined network structure of feed stocks, intermediate storage tanks, and gasoline products subject to applicable environmental standards. A mixed-integer nonlinear programming (MINLP) model is introduced which is nonconvex due to the presence of bilinear, polynomial, and fractional power terms. A mixed-integer linear programming (MILP) relaxation of the extended pooling problem is proposed and we introduce several test cases from small to medium to large scale and solve them to global optimality. The large-scale test case involves 14 feed stocks, 5 pools, and 10 products and consists of 1104 continuous variables, 150 binary variables, and 640 nonlinear terms. The nonconvexities in the large-scale case study include 410 bilinear terms, 40 polynomial terms, and 10 terms raised to a fractional power.

Eu Budget Reform Options and the Common Pool Problem

There is a consensus in the literature that the expenditure structure of the EU budget is not in line with normative criteria based e.g. on the theory of fiscal federalism. While transfer policies have a large weight, policies of a European public good type are neglected. Our contribution explains this inefficient outcome within the framework of a political-economic two-country model adapted from Besley and Coate (2003). The model's modifications integrate crucial features of the EU setting such as income heterogeneity of countries, a GNI dependent revenue formula and the existence of transfer policies in the EU budget besides both local and European public goods. We show that the common pool problem in combination with a budget cap is at the heart of overspending on transfers and under-spending on European public goods. On the base of this model we assess different reform options related to the revenue side. We show that the effects of an EU tax depend crucially on the deviation of the specific tax base's redistributive effects from income proportionality. Furthermore, our results point to the advantages of a general but limited correction mechanism.

Merged municipalities, higher debt: on free-riding and the common pool problem in politics

This paper uses the Swedish municipal amalgamation reform of 1952 to study the common pool problem in politics. The amalgams were common pools and the municipalities had incentives to free-ride on their amalgam partners by increasing debt prior to amalgamation. We find that municipalities that merged in 1952 increased their debt between 1948 and 1952 when the reform could be anticipated. The increase amounted to 52% of new debt issued or 1.5% of total revenues in the merged municipalities. But contrary to the “law of 1/n”, free-riding did not increase in common pool size.

From the Editor—Special Cluster on High-Throughput Optimization

This issue includes a Special Cluster of seven papers on the topic of High-Throughput Optimization. The concept for the cluster arose from a session that I organized on this topic for the INFORMS Annual Meeting in Pittsburgh in 2006. A call for papers for a Special Issue of the INFORMS Journal on Computing was issued in 2007, but before the due date for the papers arrived, I was appointed Editor-inChief of JOC. For this reason, the day-to-day handling of the papers devolved to a number of the JOC Area Editors: Karen Aardal, Bob Fourer, Harvey Greenberg, and John Hooker. It is these four who should be recognized as the Guest Editors for this Special Cluster. The motivation for the Special Cluster is the observation that parallel computing is no longer limited to very expensive high-end computers, making it the special preserve of well-funded agencies and companies. Most new personal computers have two or even four cores. Most computers are connected via the Internet to many other computers. The opportunity to take advantage of massive and inexpensive parallel computing is here. At the same time, there are numerous difficult and complex problems in optimization that could benefit from high-throughput computing. High-throughput optimization is the result. This is generally described as systems for solving optimization problems that require large amounts of computing resources over lengthy time periods. Bussieck, Ferris, andMeeraus open the Special Cluster with the paper “Grid-Enabled Optimization with GAMS,” which describes how the GAMS modeling system has been extended to allow optimization to take place on a loosely coupled grid of heterogeneous computing resources. Michel, See, and Van Hentenryck then describe a system for constraint programming that exploits parallelism transparently without changes to the sequential code in “Transparent Parallelization of Constraint Programming.” When a model cannot be automatically decomposed for a parallel solution, then specialized algorithms are needed for particular applications. Xu, Ralphs, Ladanyi, and Saltzman describe a framework for parallelizing branch-and-bound algorithms for solving mixed-integer programs in “Computational Experience with a Software Framework for Parallel Integer Programming.” Ferris, Maravelias, and Sundaramoorthy describe a decomposition approach for the MIP problem that arises in “Simultaneous Batching and Scheduling Using Dynamic Decomposition on a Grid.” Regis and Shoemaker describe the parallelization of a global optimization method in “Parallel Stochastic Global Optimization Using Radial Basis Functions.” Zhang, Shi, Meyer, Nazareth, and D’Souza show how a high-throughput optimization setup greatly reduces solution time in “Solving Beam-Angle Selection and Dose Optimization Simultaneously via High-Throughput Computing.” Finally, Linderoth, Margot, and Thain show how to use highthroughput optimization to improve the solutions to a gambling problem in “Improving Bounds on the Football Pool Problem by Integer Programming and High-Throughput Computing.” Solution techniques that make use of a search tree, such as branch and bound and many techniques of constraint programming, are natural candidates for parallelization. It is no surprise that six of the seven papers in the Special Cluster involve the parallelization of tree search of some kind. It is also notable that four of the papers make use of the open-source Condor framework for managing resources for highthroughput computing. The era of high-throughput optimization is here now. The hardware is cheap and accessible, and as shown in this Special Cluster, effective algorithms and software tools are available as well. I expect to see breakthroughs in the solution of some very difficult optimization problems in the next few years as these new tools come into play. This edition is rounded out by five regular papers on a variety of topics including simulation, heuristic search, and cut generation. The JOC Constraint Programming and Optimization Area is truly at the interface of operations research

Improving Bounds on the Football Pool Problem by Integer Programming and High-Throughput Computing

The football pool problem, which gets its name from a lottery-type game where participants predict the outcome of soccer matches, is to determine the smallest covering code of radius 1 of ternary words of length v. For v = 6, the optimal solution is not known. Using a combination of isomorphism pruning, subcode enumeration, and linear programming-based bounding, running on a high-throughput computational grid consisting of thousands of processors, we are able to improve the lower bound on the size of the optimal code from 65 to 71.

A common pool theory of supranational deficit ceilings

The budget deficit bias is modeled as the result of a domestic common pool problem and of an international externality. Deficits can be used to finance both unproductive and productive public spending. An optimally set supranational deficit ceiling is examined and welfare is compared to the unconstrained outcome and to the case of nationally set deficit ceilings. The supranational deficit ceiling is found to be welfare improving relative to similar national arrangements, but does not fully eliminate the deficit bias unless combined with a domestic fiscal institution allowing for precommitment to productive public spending.

The Western Expansion as a Common Pool Problem: The Contrasting Histories of the Brazilian and North American Pioneers

AbstractBefore the year 1600, Brazil and the United States were very similar regions in terms of geographic development and colonial status, and yet both countries developed in remarkably different ways. In this article, we apply institutional analysis and the common pool approach to explain differences in the western expansion of Brazil and the United States. We find that (i) such analysis complements much of the previous literature that heretofore explained differences in these regions using sociological and ideological analytical tools, and (ii) the theory that the distinctive behavior between the Brazilian bandeirantes and the North American pioneers is due to sociological factors is refuted.

Municipal Amalgamations and Common Pool Problems: The Danish Local Government Reform in 2007

Amalgamating municipalities carries risks in terms of public expenditure control. The tax base of the new amalgamated municipality represents a common pool resource. The incentive is to exploit it – that is, to spend before closing time. This article investigates last-minute spending by Danish municipalities before the local government reform in 2007 in which 271 municipalities were reduced to 98. It shows that local councillors exploited this situation to finance local projects before closing time. The article thus demonstrates the occurrence of common pool problems in amalgamation situations. However, it also shows that it is the availability of a common pool that matters, not its size.

Computational Comparison of Piecewise−Linear Relaxations for Pooling Problems

This work discusses alternative relaxation schemes for the pooling problem, a theoretically and practically interesting optimization problem. The problem nonconvexities appear in the form of bilinear terms and can be addressed with the relaxation technique based on the bilinear convex and concave envelopes. We explore ways to improve the relaxation tightness, and thus the efficiency of a global optimization algorithm, by employing a piecewise linearization scheme that partitions the original domain of the variables involved and applies the principles of bilinear relaxation for each one of the resulting subdomains. We employ 15 different piecewise relaxation schemes with mixed-integer representations and conduct a comprehensive computational comparison study over a collection of benchmark pooling problems. For each case, various partitioning variants can be envisioned, cumulatively accounting for a total of 56 700 relaxations. The results demonstrate that some of the schemes are clearly superior to their c...

Merged Municipalities, Higher Debt: On Free-Riding and the Common Pool Problem in Politics

This paper uses the Swedish municipal amalgamation reform of 1952 to study the common pool problem in politics. The amalgams were common pools and the municipalities had incentives to free-ride on their amalgam partners by increasing debt prior to amalgamation. We find that municipalities that merged in 1952 increased their debt between 1948 and 1952 when the reform could be anticipated. The increase amounted to 52% of new debt issued or 1.5% of total revenues in the merged municipalities. But contrary to the “law of 1/n”, free-riding did not increase in common pool size.

A new Lagrangean approach to the pooling problem

We present a new Lagrangean approach for the pooling problem. The relaxation targets all nonlinear constraints, and results in a Lagrangean subproblem with a nonlinear objective function and linear constraints, that is reformulated as a linear mixed integer program. Besides being used to generate lower bounds, the subproblem solutions are exploited within Lagrangean heuristics to find feasible solutions. Valid cuts, derived from bilinear terms, are added to the subproblem to strengthen the Lagrangean bound and improve the quality of feasible solutions. The procedure is tested on a benchmark set of fifteen problems from the literature. The proposed bounds are found to outperform or equal earlier bounds from the literature on 14 out of 15 tested problems. Similarly, the Lagrangean heuristics outperform the VNS and MALT heuristics on 4 instances. Furthermore, the Lagrangean lower bound is equal to the global optimum for nine problems, and on average is 2.1% from the optimum. The Lagrangean heuristics, on the other hand, find the global solution for ten problems and on average are 0.043% from the optimum.

The Fishery as a Watery Commons: Lessons from the Experiences of Other Public Policy Areas for U.S. Fisheries Policy

Open access, combined with modern technologies of fishing, has created serious problems of overfishing and threatens the sustainability of many U.S. fisheries. The common pool problem - the ocean version of the tragedy of the commons - is the root cause of the overfishing. The major regulatory policies of the past few decades that have tried to address overfishing - restrictions on fishing methods and inputs (in essence, command and control regulation) - have largely been failures. Indeed, they have often perversely exacerbated fisheries' overfishing problems by encouraging fishing derbies or races for the fish. Fisheries are not alone in facing a common pool problem. Other areas of the U.S. economy have confronted similar problems, and public policies have developed to deal with them. This paper discusses seven of these other areas: the use of the electromagnetic spectrum, the control of sulfur dioxide emissions by electric utilities, grazing on public lands, forest logging on public lands, oil-gas-coal extraction from public lands and offshore waters, hard rock mineral (metal) mining, and surface water usage. Important lessons can be gleaned from the policies that have been developed in these other areas, and this paper applies those lessons to the design of U.S. fisheries policy.

Legislative Representation, Bargaining Power and the Distribution of Federal Funds: Evidence from the US Congress

This article investigates the relationship between representation in legislatures and the geographic distribution of federal funds. In a legislative bargaining model, we demonstrate that funds are concentrated in high representation areas, and two channels underlie this result. The proposal power channel reflects the role of representation in committee assignments, and the vote cost channel reflects the role of representation in coalition formation. In our empirical analysis, we find that small states, relative to large states, receive more funding in the US Senate, relative to the House. We also find empirical support for the two channels underlying this relationship. Throughout the history of federations and international organisations, a key consti tutional design question involves how jurisdictions of different sizes are represented in legislative bodies and how their votes will ultimately be weighted. Large jurisdictions typically argue for delegation sizes or voting weights that are roughly proportional to population whereas small jurisdictions typically argue for an equal number of delegates or votes for each jurisdiction. This conflict over representation is presumably driven, at least in part, by a belief that increased representation strengthens bargaining power in the battle over the geographic distribution of funds authorised by these legislative bodies. In this article, I analyse the relationship between legislative representation, bar gaining power and the geographic distribution of federal funds from both a theoretical and empirical perspective. In addition to investigating the existence of a relationship between representation and federal funds, I also explore the underlying mechanisms. That is, I ask which political institutions, such as agenda control provided to commit tees and the formation of coalitions under majority rule, shape the bargaining power of jurisdictions in central legislatures and thus facilitate the relationship between repre sentation and the distribution of federal spending. To provide a framework for analysing these issues, I begin by developing a simple theoretical model of a central legislature that determines the crossjurisdiction distri bution of local public goods, which are financed by a national tax base. Both delegation sizes and population are variable, allowing me to consider a variety of institutional arrangements for jurisdictional representation. Legislators represent local interests, and, given the common pool problem associated with concentrated benefits and dis persed tax costs, attempt to increase ownjurisdiction spending but to restrain spending elsewhere. The model follows a recent literature on legislative bargaining, in which a proposer must secure a majority of votes in order to pass legislation and, due to the

BAILOUTS IN FEDERATIONS: IS A HARD BUDGET CONSTRAINT ALWAYS BEST?*

This article analyses hard and soft budget constraints in a federation, where there is a moral hazard problem between the central and the regional governments. Regional governments can avoid a bailout from the center by exerting costly effort. In this setting, a hard budget constraint is not always optimal because it can provide excessive incentives for high effort, and thus discourage investment that is socially efficient. Thus, a hard budget constraint can imply the opposite kind of inefficiency that emerges under a soft budget constraint, where the common pool problem can give rise to inefficiently low effort and overinvestment.