E. Andrew Boyd

Efficient operation of natural gas transmission systems: a network-based heuristic for cyclic structures

In this paper we propose a heuristic solution procedure for fuel cost minimization on gas transmission systems with a cyclic network topology, that is, networks with at least one cycle containing two or more compressor station arcs. Our heuristic solution methodology is based on a two-stage iterative procedure. In a particular iteration, at a first stage, gas flow variables are fixed and optimal pressure variables are found via dynamic programming. At a second stage, pressure variables are fixed and an attempt is made to find a set of flow variables that improve the objective function by exploiting the underlying network structure. Empirical evidence supports the effectiveness of the proposed procedure outperforming the best existing approach to the best of our knowledge.

Fenchel Cutting Planes for Integer Programs

A technique for generating cutting planes for integer programs is introduced that is based on the ability to optimize a linear function on a polyhedron rather than explicit knowledge of the underlying polyhedral structure of the integer program. The theoretical properties of the cuts and their relationship to Lagrangian relaxation are discussed, the cut generation procedure is described, and computational results are presented.

A Reduction Technique for Natural Gas Transmission Network Optimization Problems

We address the problem of minimizing the fuel consumption incurred by compressor stations in steady-state natural gas transmission networks. In the practical world, these type of instances are very large, in terms of the number of decision variables and the number of constraints, and very complex due to the presence of non-linearity and non-convexity in both the set of feasible solutions and the objective function. In this paper we present a study of the properties of gas pipeline networks, and exploit them to develop a technique that can be used to reduce significantly problem dimension, without disrupting problem structure, making it more amenable to solution.

Heuristics for Minimum Cost Steady-State Gas Transmission Networks

In this paper we propose and present two heuristics for the problem of minimizing fuel cost on steady-state gas transmission problems on looped networks. One of the procedures is based on a two-stage iterative procedure, where, in a given iteration, gas flow variables are fixed and optimal pressure variables are found via dynamic programming in the first stage. In the second stage, the pressure variables are fixed and an attempt is made to find a set of flow variables that improve the objective function by exploiting the underlying network structure. The other proposed heuristic adapts some concepts from generalized reduced gradient methods to attempt to find the direction step. This work focuses on looped network topologies, that is, networks with at least one cycle containing two or more compressor stations. These topologies possess the highest degree of difficulty in real-world problems.

Cutting planes for mixed-integer knapsack polyhedra

An algorithm for generating cutting planes for mixed-integer knapsack polyhedra is described. The algorithm represents an exact separation procedure and is based on a general methodology proposed by one of the authors in an earlier paper. Computational results are presented.

An Efficient Algorithm for Solving An Air Traffic Management Model of the National Airspace System

Growth in traffic and changes in traffic patterns have caused an increase in the congestion and delay in the National Airspace System. Air traffic delay is very costly to the airlines, and minimizing this delay has been a subject of research for over a decade. A large integer programming model developed at the MITRE Corporation for minimizing air traffic delay is presented. Solving problem instances arising from this model involves the use of preprocessing, constraint strengthening, and a carefully designed computer implementation. Results are presented, demonstrating that the model can be solved to provable optimality in real time for problem instances involving over 1 million binary variables.

Solving Large Integer Programs Arising from Air Traffic Flow Problems

Recent integer programming techniques are used to solve problems arising in the control of national air traffic. Two large problems based on real data and developed by the MITRE Corporation for the Federal Aviation Administration are solved to provable optimality. The work was motivated by the inability to solve these problems using commercial integer programming codes. The results demonstrate that, through effective use of recent integer programming techniques, air traffic control problems can be solved that could not be solved without these techniques.

Solving 0/1 integer programs with enumeration cutting planes

A cutting plane technique with applicability to the solution of integer programs is presented. The computational value of this technique is demonstrated by applying it to a collection of seven difficult integer programs arising from real-world applications. Four of the seven problems are solved to optimality without the aid of branch and bound, and six of the seven problems have the gap between the value of the integer program and its linear programming relaxation closed by over 98%.

A fully polynomial epsilon approximation cutting plane algorithm for solving combinatorial linear programs containing a sufficiently large ball

A cutting plane algorithm is presented for finding @e-approximate solutions to integer programs contained in the unit hypercube and represented by a separation oracle. Under the assumption that a polynomially bounded ball is contained in the feasible region of the problem, it is demonstrated that the algorithm is an oracle fully polynomial @e approximation scheme. Implications of the result for 0/1 integer programming are discussed.

Resolving degeneracy in combinatorial linear programs: steepest edge, steepest ascent, and parametric ascent

While variants of the steepest edge pivot rule are commonly used in linear programming codes they are not known to have the theoretically attractive property of avoiding an infinite sequence of pivots at points of degeneracy. An example is presented demonstrating that the steepest edge pivot rule can fail to terminate finitely. It is then shown that a natural extension of the steepest edge pivot rule based on steepest ascent is guaranteed to determine a direction of ascent or a proof that no such direction exists after a finite number of pivots, although without modification the extension may not terminate with an ascent direction corresponding to an edge. Finally, it is demonstrated that a computationally more efficient pivot rule proposed by Magnanti and Orlin has similar theoretical properties to steepest ascent with probability 1independent of the linear program being solved. Unlike alternative methods such as primal lexicographic rules and Blands rule, the algorithms described here have the advantage that they choose the pivot element without explicit knowledge of the set of all active constraints at a point of degeneracy, thus making them attractive in combinatorial settings where the linear program is represented implicitly.