R-3 Repository :: Browsing by Author "Bixby, Robert E."

Browsing by Author "Bixby, Robert E."

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A Short Proof of a Decomposition Theorem for Max-Flow Min-Cut Matroids
(1986-11) Bixby, Robert E.; Rajan, Arvind
This report contains short proofs of two known matroid decomposition results, both of which are based on a decomposition algorithm of Truemper. The main result is a recent theorem of Truemper and Tseng for the class of matroids with the max-flow min-cut property, a class characterized by Seymour. The theorem says essentially that every matroid in this class is either isomorphic to F tau or is decomposable into a 3-sum in a well-defined way. The second result describes the structure of regular matroids, and is an important ingredient in Seymour's decomposition theorem for this class.
A Short Proof of the Truemper-Tseng Theorem on Max-Flow Min-Cut Matroids
(1987-10) Bixby, Robert E.; Rajan, Arvind
Seymour has characterized the matroids satisfying the integral max-flow min-cut property with respect to a fixed element. Truemper and Tseng subsequently proved a decomposition theorem for this class, similar in spirit to Wagner's characterization of the regular (totally unimodular) matroids. The purpose of this paper is to give a short, self-contained exposition of the Truemper-Tseng result.
A subgradient algorithm for nonlinear integer programming and its parallel implementation
(1991) Wu, Zhijun; Dennis, John E., Jr.; Bixby, Robert E.
This work concerns efficiently solving a class of nonlinear integer programming problems: min $\{f(x)$: $x \in \{0,1\}\sp{n}\}$ where $f(x)$ is a general nonlinear function. The notion of subgradient for the objective function is introduced. A necessary and sufficient condition for the optimal solution is constructed. And a new algorithm, called the subgradient algorithm, is developed. The algorithm is an iterative procedure, searching for the solution iteratively among feasible points, and in each iteration, generating the next iterative point by solving the problem for a local piecewise linear model of the original problem which is constructed with supporting planes for the objective function at a set of feasible points. Special continuous optimization techniques are used to compute the supporting planes. The problem for each local piecewise linear model is solved by solving an equivalent linear integer program. The fundamental theory for the new approach is built and all related mathematical proofs and derivations such as proofs for convergence properties, the finiteness of the algorithm, as well as the correct formulation of the subproblems are presented. The algorithm is parallelized and implemented on parallel distributed-memory machines. The preliminary numerical results show that the algorithm can solve test problems effectively. To implement the subgradient algorithm, a parallel software system written in EXPRESS C is developed. The system contains a group of parallel subroutines that can be used for either continuous or discrete optimization such as subroutines for QR, LU and Cholesky factorizations, triangular system solvers and so on. A sequential implementation of the simplex algorithm for linear programming also is included. Especially, a parallel branch-and-bound procedure is developed. Different from directly parallelizing the sequential binary branch-and-bound algorithm, a parallel strategy with multiple branching is used for good processor scheduling. Performance results of the system on NCUBE are given.
A Test Set of Real-World Mixed Integer Programming Problems
(1991-11) Bixby, Robert E.; Boyd, E. Andrew; Indovina, Ronni R.
An Almost Linear-Time Algorithm for Graph Realization
(1985-03) Bixby, Robert E.; Wagner, Donald K.
Given a {0,1}-matrix M, the graph realization problem for M is to find a tree such that the columns of M are incidence vectors of paths in T, or to show that no such T exists. An algorithm is presented for this problem the time complexity of which is very nearly linear in the number of ones in M.
An efficient simplex-based method for solving large linear programs
(1995) Dadmehr, Shireen Sara; Bixby, Robert E.
A simplex-based method of solving specific classes of large-scale linear programs is presented. The structure of both staircase and block-angular linear programs is exploited to construct an advanced or "crash" basis to be used with a simplex-based linear program solver. First, the constraint matrix is decomposed into blocks. In contrast to many other systems that require additional information about the form of the linear program, the method described here determines this decomposition without prior knowledge of the matrix structure. As the problem of automatically finding such a decomposition is NP-complete, a heuristic is used to discover blocks within the constraint matrix. After a decomposition of the constraint matrix is determined, smaller linear programs called subproblems are formed. These subproblems are solved using a simplex-based solver, and the solution information is used to construct an advanced or "crash" basis for the original linear program. In contrast to decomposition methods that iteratively solve subproblems to obtain a solution to the original problem, this approach requires solving each subproblem at most twice. Finally, the original linear program is solved by providing the advanced basis to a simplex-based solver. Results indicate that this method solves some large linear programs much faster than state-of-the-art simplex-based solvers do. For example, for a set of linear programs that range in size from 5,000 x 9,000 to 30,000 x 50,000 constraints and variables, the presented method solves each linear program in about a tenth the number of simplex iterations. This reduces the total time required to solve each linear program by a factor of from two to five.
Finding Embedded Network Rows in Linear Programs I: Extraction Heuristics
(1986-08) Bixby, Robert E.; Fourer, Robert
An embedded network within a linear program is, roughly speaking, a subset of constraints that represent conservation of flow. In this paper, we examine three broad classes of heuristic techniques - row-scanning deletion, column-scanning deletion, and row-scanning addition - for the extraction of large embedded networks. We describe a variety of implementations, and compare their performance on varied test problems. The success of our tests depends, in part, on several preprocessing steps that scale the constraint matrix and that set aside certain rows and columns. Efficiency of the subsequent network extraction is dependent on the implementation, in predictable ways. Effectiveness is harder to explain; the more sophisticated and expensive implementations seem to be more reliable, but much simpler implementations sometimes find equally large networks. The largest networks are obtained by applying a final augmentation phase, which is studied in the second part of this paper.
Implementing the Simplex Method: The Initial Basis
(1990-10) Bixby, Robert E.
Matroid Optimization and Algorithms
(1990-06) Bixby, Robert E.; Cunningham, William H.
This paper reviews matroid optimization and algorithms including applications of matroid intersection; submodular functions and polymatroids; submodular flows and other general models; matroid connectivity algorithms; recognition of representability; and matroid flows and linear programming.
Notes on Combinatorial Optimization
(1987-10) Bixby, Robert E.
On the Integrality Gap of the Subtour Relaxation of the Traveling Salesman Problem for Certain Fractional 2-matching Costs
(2014-04-11) Fast, Caleb; Hicks, Illya V.; Bixby, Robert E.; Cooper, Keith D.; Tapia, Richard A.
This thesis provides new bounds on the strength of the subtour relaxation of the Traveling Salesman Problem (TSP) for fractional 2-matching cost instances whose support graphs have no fractional cycles larger than five vertices. This work provides insight for improving approximation heuristics for the TSP and into the structure of solutions produced by the subtour relaxation. Guided by a T-join derived from the subtour relaxation, I form a tour by adding edges to the subtour relaxation. By this constructive process, I prove that the optimal solution of the TSP is within 4/3, 17/12, or strictly less than 3/2 of the optimal solution of the subtour relaxation. Thus, this thesis takes a step towards proving the 4/3 conjecture for the TSP and the development of a 4/3 approximation algorithm for the TSP. These developments would provide improved approximations for applications such as DNA sequencing, route planning, and circuit board testing.
Optimizing over the cut cone: A new polyhedral algorithm for the maximum-weight cut problem
(1991) Saigal, Sanjay; Bixby, Robert E.
Polyhedral cutting-plane algorithms for hard combinatorial problems have scored notable successes. However, computational research on the Maximum-Weight Cut Problem (MCP) on undirected graphs has been inconclusive. In 1988, Barahona suggested a new polyhedral algorithm that, given a good initial solution, attempts to prove optimality. If the initial cut is non-optimal, it is iteratively improved until optimal. The expected advantages are three-fold. If a good, fast heuristic is used, an optimal solution may be available. The algorithm can then prove optimality fast. Secondly, if time is a serious constraint, prematurely terminating the algorithm yields a cut at least as good as the original. Finally, since the algorithm nominally optimizes over the cut cone rather than the cut polytope, the underlying separation problem is very simple. This research explores Barahona's algorithm on a class of MCP instances arising in statistical mechanics. The graphs are toroidal grids, together with an additional universal vertex. By considering different integer programming formulations, it has been possible to design a fast algorithm that replaces optimization over the cut polytope by repeated optimization over the intersection of the cut cone and the unit cube. This latter polyhedron is shown to be equivalent to the multicut polytope, and its basic facet classes are identified. The final algorithm is successful in solving MCP instances over 70 x 70 grids, over 5 times bigger than previous algorithms. Substantial improvements in computation time have also been achieved.
Progress in Linear Programming
(1993-09) Bixby, Robert E.
There is little doubt that barrier methods are now indispensable tools in the solution of large-scale linear programming problems. However, it is our opinion that the results of Lustig, Marsten and Shanno (hereafter LMS) somewhat overstate the performance of these methods relative to the simplex method. We will present a sightly different view of progress in linear programming, one in which barrier methods do not dominate in the solution of large-scale problems.
Recovering an Optimal LP Basis from an Interior Point Solution
(1991-10) Bixby, Robert E.; Saltzman, Matthew J.
An important issue in the implementation of interior point algorithms for linear programming is the recovery of an optimal basic solution from an optimal interior point solution. In this paper we describe a method for recovering such a solution. Our implementation links a high-performance interior point code (OB1) with a high-performance simplex code (CPLEX). Results for our computational tests indicate that basis recovery can be done quickly and efficiently.
Reoptimization in interior-point methods with application to integer programming
(1999) McZeal, Cassandra Moore; Bixby, Robert E.; Zhang, Yin
This thesis examines current reoptimization techniques for interior-point methods available in the literature and studies their efficacy in a branch-and-bound framework for 0/1 mixed integer programming problems. This work is motivated by the observation that there are instances of integer programming problems where each individual linear program generated in a branch-and-bound tree can be solved much faster by an interior-point algorithm than by a simplex algorithm, in spite of the fact that effective "warm-start" techniques are available for the latter but not for the former. Because of many unresolved issues surrounding effective reoptimization techniques for interior-point methods, interior-point algorithms have not been commonly used as linear programming solvers in a branch-and-bound framework. In this work, we identify and examine a number of key factors that may affect and even preclude effective reoptimization for interior-point algorithms in the branch-and-bound framework, including change in optimal partition, distance to optimality, and primal infeasibility. We conclude that even though various "warm-start" techniques are capable of reducing the reoptimization cost to some extent, for certain problem instances a rapid reoptimization can not always be expected from interior-point methods due to their inherent limitations. Continued research is needed in the direction of the present study in order to provide comprehensive guidelines for the most effective utilization of interior-point algorithms in a branch-and-bound algorithm.
Solving a Truck Dispatching Scheduling Problem Using Branch-and-Cut
(1993-09) Bixby, Robert E.; Lee, Eva K.
A branch-and-cut IP solver is developed for a class of structured 0/1 integer programs arising from a truck dispatching scheduling problem. This problem involves a special class of knapsack equality constraints. Families of facets for the polytopes associated with individual knapsack constraints are identified, and in some cases, a complete characterization of a polytope is obtained. In addition, a notion of "conflict graph" is utilized to obtain an approximating node-packing polytope for the convex hull of all 0/1 solutions. The branch-and-cut solver generates cuts based on both the knapsack equality constraints and the approximating node-packing polytope, and incorporates these cuts into a tree-search algorithm that uses problem reformulation and linear programming-based heuristics at each node in the search tree to assist in the solution process. Numerical experiments are performed on large-scale real instances supplied by Texaco Trading &Transportation, Inc. The optimal schedules correspond to cost savings for the company and greater job satisfaction for drivers due to more balanced work schedules and income distribution.
Solving Real-World Linear Programs: A Decade and More of Progress
(2001-10) Bixby, Robert E.
This paper is an invited contribution to the 50th anniversary issue of the journal Operations Research, published by the Institute for Operations Research and the Management Sciences (INFORMS). It describes one person's perspective on the development of computational tools for linear programming. The paper begins with a short, personal history, followed by historical remarks covering the some 40 years of linear-programming developments that predate my own involvement in this subject. It concludes with a more detailed look at the evolution of computational linear programming since 1987.
Solving structured 0/1 integer programs arising from truck dispatching scheduling problems
(1993) Lee, Eva Kwok-Yin; Bixby, Robert E.
A branch-and-cut IP solver is developed for a class of structured 0/1 integer programs arising from a truck dispatching scheduling problem. This problem is characterized by a group of set partitioning constraints and a group of knapsack equality constraints of a specific form. Families of facets for the polytopes associated with individual knapsack constraints are identified, and in some cases, a complete characterization of a polytope is obtained. In addition, a notion of "conflict graph" is introduced and utilized to obtain an approximating node-packing polytope for the convex hull of all 0/1 solutions. The branch-and-cut solver generates cuts based on both the knapsack constraints and the approximating node-packing polytope, and incorporates these cuts into a tree-search algorithm that uses problem reformulation and linear programming-based heuristics at each node in the search tree to assist in the solution process. Numerical experiments are performed on large-scale real instances supplied by Texaco Trading & Transportation, Inc. The optimal schedules obtained correspond to cost savings for the company and greater job satisfaction for drivers due to more balanced work schedules and income distribution. It is noteworthy that this is apparently the first time that branch-and-cut has been applied to an equality constrained problem in which the entries in the constraint matrix and right hand side are not purely 0/1.
The solution of a class of limited diversification portfolio selection problems
(1997) Butera, Gwyneth Owens; Bixby, Robert E.; Dennis, John E., Jr.
A branch-and-bound algorithm for the solution of a class of mixed-integer nonlinear programming problems arising from the field of investment portfolio selection is presented. The problems in this class are characterized by the inclusion of the fixed transaction costs associated with each asset, a constraint that explicitly limits the number of distinct assets in the selected portfolio, or both. Modeling either of these forms of limiting the cost of owning an investment portfolio involves the introduction of binary variables, resulting in a mathematical programming problem that has a nonconvex feasible set. Two objective functions are examined in this thesis; the first is a positive definite quadratic function which is commonly used in the selection of investment portfolios. The second is a convex function that is not continuously differentiable; this objective function, although not as popular as the first, is, in many cases, a more appropriate objective function. To take advantage of the structure of the model, the branch-and-bound algorithm is not applied in the standard fashion; instead, we generalize the implicit branch-and-bound algorithm introduced by Bienstock (3). This branch-and-bound algorithm adopts many of the standard techniques from mixed-integer linear programming, including heuristics for finding feasible points and cutting planes. Implicit branch-and-bound involves the solution of a sequence of subproblems of the original problem, and thus it is necessary to be able to solve these subproblems efficiently. For each of the two objective functions, we develop an algorithm for solving its corresponding subproblems; these algorithms exploit the structure of the constraints and the objective function, simplifying the solution of the resulting linear systems. Convergence for each algorithm is proven. Results are provided for computational experiments performed on investment portfolio selection problems for which the cardinality of the universe of assets available for inclusion in the selected portfolio ranges in size from 52 to 1140.
Very Large-Scale Linear Programming: A Case Study in Combining Interior Point and Simplex Methods
(1991-05) Bixby, Robert E.; Gregory, John W.; Lustig, Irvin J.; Marsten, Roy E.; Shanno, David F.
Experience with solving a 12,753,313 variable linear program is described. This problem is the linear programming relaxation of a set partitioning problem arising from an airline crew scheduling application. A scheme is described that requires successive solutions of small subproblems, yielding a procedure that has little growth in solution time in terms of the number of variables. Experience using the simplex method as implemented in CPLEX, an interior point method as implemented in OB1, and hybrid interior point/simplex approach is reported. The resulting procedure illustrates the power of an interior point/simplex combination for solving very large-scale linear programs.

Browsing by Author "Bixby, Robert E."

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