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We present a bounding technique for use in implicit enumeration algorithms for solving the integer linear programming problem with binary variables. The main assumptions used by this technique are the associated linear program, obtained by dropping the integrality constraints on the variables, possesses a unique optimal solution, this optimal solution is not binary, and a good feasible solution to the original problem is available. An alternative to the last assumption which is weaker is also presented. We show that joint bounds can be obtained on the values of a subset of the variables. In addition we given an efficient method to implement this bounding technique. Finally, a class of problems particularly well-suited to this bounding procedure is specified. (Author)

We introduce a general technique to create an extended formulation of a mixed-integer program. We classify the integer variables into blocks, each of which generates a finite set of vector values. The extended formulation is constructed by creating a new binary variable for each generated value. Initial experiments show that the extended formulation can have a more compact complete description than the original formulation. We prove that, using this reformulation technique, the facet description decomposes into one ``linking polyhedron'' per block and the ``aggregated polyhedron''. Each of these polyhedra can be analyzed separately. For the case of identical coefficients in a block, we provide a complete description of the linking polyhedron and a polynomial-time separation algorithm. Applied to the knapsack with a fixed number of distinct coefficients, this theorem provides a complete description in an extended space with a polynomial number of variables.

There were several accomplishments of this research, both theoretical and computational. In joint work with Todd, we presented a cutting plane primal projective interior point method which we applied to matching problems, with encouraging computational results. Primal projective methods require a method to update the dual; we showed how various dual updates are related to each other and we also derived a dual projective algorithm. We derived a polynomial-time shifted barrier warm start algorithm which can be used in a cutting plane method; we showed that the directions obtained are strongly related to the directions derived in the work with Todd; computational results showed that the algorithm can be useful in some situations. The grant partially supported a Ph. D. student, Brian Borchers, who received his degree in August, 1992. His thesis concerned the use of branch-and-bound methods and contained good computational results as well as interesting theoretical observations. One paper from this thesis describes how the primal-dual interior point method can be used efficiently in a branch-and-bound method for solving mixed integer linear programming problem. Another paper describes how branch and bound algorithms for nonlinear integer programming problems can be improved. Borchers and I also developed a primal-dual interior point cutting plane method for solving linear ordering problems; the computational results for this algorithm were very encouraging, with run times comparable to those required by a simplex based cutting plane algorithm.

This report examines the role played by several kinds of ceiling points in solving the pure, general integer linear programming problem (ILP). While no assumptions are made concerning the structure or signs of the data of the problem, it is assumed that the feasible region for (ILP) is non-empty and bounded. A ceiling point with respect to a single constraint maybe thought of as an integer solution on or close to the boundary of the feasible region defined by the constraint. The definition of a ceiling point with respect to a single constraint is extended to take multiple constraints into consideration simultaneously, defining what is called a feasible ceiling point. It is shown that the set all feasible ceiling points contains at least one optimal solution for (ILP). A related class of solutions called feasible 1-ceiling points is also characterized and shown to contain all optimal solutions for (ILP). Moreover, 1- ceiling points are computationally easier to identify than ordinary ceiling points and may be sought with respect to one constant at a time. It is also demonstrated that solving (ILP) requires only enumerating feasible 1-ceiling points with respect to a subset of all functional constraints. Keywords: Integer variables; Enumeration algorithms.

The Army is reducing and reshaping its force structure to adapt to the nation's changing defense needs and budget constraints. In response to these changes, Army Materiel Command (AMC) will submit facility realignment and closure recommendations in FY93 and FY95. This thesis develops a bi-criteria mixed integer programming model with the objectives of minimizing operating costs and maximizing a measure of military value to assist AMC in the generation of alternative realignments. Realignment of depot maintenance, research and development, test and evaluation, and administrative functions are considered on 32 AMC installations. An extensive empirical study demonstrates the applicability of the developed approach.

This report develops a hybrid algorithm to solve the binary integer linear programming problem. This problem involves optimizing a linear objective function subject to a set of linear inequalities where, in addition, we require the variables to take on the value 0 or 1. the approach developed is that of implicit enumeration; that is, the set of all possible binary combinations of values for the variables is searched without considering each combination explicitly. This is accomplished by applying a series of tests which when satisfied allow immediate elimination of large subsets of these completions. It is in the choice of tests to be used that this algorithm may be termed a hybrid. Borrowing penalties and pseudocosts as well as binary infeasibility and conditional binary infeasibility tests from previous approaches, the algorithm is built to use each of their strengths. In addition, an existing heuristic procedure is used to generate a good feasible binary point at the outset. Thus, a good initial bound on the optimal function value is available. (Author)

The motivation for this work has been the need for a practical procedure to solve the maximum-weight cut problem (MCP) in undirected graphs. Our primary focus has been on problems arising from considerations in statistical mechanics. These problems are typically posed on grid graphs and some natural variants. There has been significant progress in two areas: solution of the maximum-weight cut problem and development of simplex-based tools for integer programming. Codes developed have been widely used to improve solution time.

Presented is an integer-programming approach to item pool design that can be used to calculate an optimal blueprint for an item pool to support an existing testing program. The results are optimal in the sense that they minimize the efforts involved in actually producing the items as revealed by current item writing patterns. Also presented is an adaptation of the models for use as a set of monitoring tools in item pool management. The approach is demonstrated empirically for an item pool designed for the Law School Admission Test. (Contains 2 tables and 30 references.) (Author/SLD)

The limited success of exact algorithms for solving integer programming problems has encouraged the development of heuristic procedures for efficiently obtaining solutions that are at least close to optimal. This document presents three heuristic procedures for 0-1 integer programming problems having only inequality constraints. These procedures are based on Hillier's previous heuristic procedures for general integer linear programming. All three were successfully run on problems with up to 500 variables with only modest execution times. The quality of the solutions for these problems were, in general, very good and often were optimal. When the best of the solutions obtained by the three procedures was taken, the final solution was optimal for 24 of 45 randomly generated problems. These procedures can be used for problems that are too large to be computationally feasible for exact algorithms. In addition, they can be useful for smaller problems by quickly providing an advanced starting solution for an exact algorithm.

The report presents a synthesis of the Balasian implicit enumeration approach to integer linear programming with the approach typified by Land and Doig and by Roy, Bertier, and Nghiem. This synthesis results from the use of an imbedded linear program to compute surrogate constraints that are as 'strong' as possible in a sense slightly different from that originally used by Glover. A very simple implicit enumeration algorithm fitted with optional imbedded linear programming machinery was implemented and tested extensively on an IBM 7044 computer. Use of the imbedded linear program dramatically reduced solution time in virtually every case and sufficied to render the tested algorithm superior to the other five implicit enumeration algorithms for which comparable published experience was available. Existing evidence suggests that the present approach should permit the routine solution of practical integer problems involving hundreds of variables.

The paper presents a generalization, called polaroid, of the concept of polar sets. A list of properties satisfied by polaroids is established indicating that the new concept may be fruitfully used in an area of non-convex (called here polar) programming as well as in integer programming, by means of polaroid cuts; this class of new cuts contains the ones defined by Tuy for concave programming (a special case of polar programming) and by Balas for integer programming; it furthermore provides for new degrees of freedom in the construction of algorithms in the above-mentioned areas of mathematical programming.

Contents: Resource Decomposition: A Conceptual Outline; Branching in the Rudimentary Branch-and-Bound Algorithm; Computing Bounds: Single Linking Constraint Case; Computing Bounds: Multiple Linking Constraints Case; Computational Results; and Conclusions, Extensions, and Areas for Future Research.

Fundamental Disjunctive Conic Cut (DCC) methodology for Mixed-Integer Conic Linear Optimization (MICO) was developed. To describe the convex hull of the intersection of a convex set E and a linear disjunction is the fundamental problem, and that served as the core of solution techniques for MICO. It was proved that if there exists a cone K that has the same intersection with the boundary of the disjunction as the convex set E, then the convex hull of the disjunction is the intersection of E with K. While uniqueness of a DCC is proved for general MICO, the existence of such a cone is difficult to prove for the general case. Thorough analysis of a parametric family of quadrics allows to prove the existence and uniqueness of a second order cone, when E is the intersection of an affine space and a second order cone. An efficiently computable method was developed for finding that cone, which provided novel and powerful DCCs for Mixed Integer Second Order Cone Optimization (MISOCO), which can be used in branch-and-cut algorithms when solving MISOCO problems. All special and degenerate cases are carefully analyzed and easy to compute criteria are developed to compute a DCC for all cases. Limited, but rigorous computational experiments gave strong indication of the power of the DCCs.

The Marine Corps has long been successful in assigning its available personnel to vacant billets. However, by our research, the Marine Corps has not been as successful in minimizing assignment costs when moving a Marine to another permanent duty station. With increased pressure on cost savings due to shrinking budgets, the importance of cost minimizing efforts is becoming more significant. This thesis examines the Marine Corps personnel assignment process and proposes a methodology for optimizing the allocation of Marine Corps personnel that minimizes assignment costs while taking into account constraints such as military occupational specialty, billet vacancies, duty station preference, and seniority. Optimization is achieved by incorporating an integer programming model into the personnel assignment process. The model is tested by contrasting the results of the actual assignments of a 15-Marine sample with the results of simulated optimization assignments of the same sample. The findings of this thesis show that the proposed methodology is both valid and feasible, and that it could yield significant monetary savings for the Marine Corps.

A primal feasible (all-integer) integer linear programming algorithm has been developed and programmed, together with a related procedure for obtaining a first feasible solution. Once a feasible solution is found, the algorithm maintains feasibility at each stage, in contrast to other algorithms that have been programmed and are currently available. These other algorithms do not achieve feasibility until the optimal solution is reached. The primal feasible algorithm is based on a particular way of applying the cutting planes previously developed by R. E. GOMORY, and on a specific interpretation of their role. The finiteness of convergence has been established for two-dimensional problems but not for the general case; however, there appears to be at least computational convergence in a considerable fraction of the cases. In addition, a Generalized Euclidean Algorithm for finding the greatest common divisor for more than two numbers is defined. The solution of systems of linear diophantine equations is presented in terms of integer linear programming. Some geometric considerations that help to illuminate the workings of the algorithm, are examined.

Detection of an Unmanned Air Vehicle by radar is dependent on many variables including range, altitude, and relative orientation. Given a radar location and appropriate model for the likelihood of detection, a path plan can be created for an Unmanned Air Vehicle which constrains the probability of detection. In this paper such an approach is taken using a linearized detection model. The detection model and the Unmanned Air Vehicle's dynamics are represented as a linear program subject to mixed integer constraints. This mixed integer linear program (MILP) is then solved with commercial software which has been traditionally used by the Operations Research community. This approach searches for all feasible solutions and produces the best path plan based on the user specified parameters.

Special Operations Forces (SOF) are frequently employed to conduct missions not within the limited boundaries of unilateral special operations. These operations often involve cooperation with general purpose forces (GPF) and are often under their command. This thesis argues that these cooperative efforts are better examined as integrated operations rather than special or conventional operations. These operations require SOF to conduct specialized tasks facilitating the introduction of follow-on GPF to complete the mission. This thesis develops a theory of integrated operations by examining six operations, previously considered under conventional wisdom as either special or conventional. It rejects much of the myth which surrounds these operations and offers a revisionist interpretation of the necessary and sufficient conditions for success in these endeavors. The thesis then goes on to compare cases of these special units using organizational theory to determine the sources of integration. The cases examined indicate a causal relationship between organizational factors related to command and training and a special units ability to integrate with GPF. The thesis concludes by recommending the realignment of select SOF in the current force structure to better meet the challenges of future integrated operations.

Special Operations Forces (SOF) are frequently employed to conduct missions not within the limited boundaries of unilateral special operations. These operations often involve cooperation with general purpose forces (GPF) and are often under their command. This thesis argues that these cooperative efforts are better examined as integrated operations rather than special or conventional operations. These operations require SOF to conduct specialized tasks facilitating the introduction of follow-on GPF to complete the mission. This thesis develops a theory of integrated operations by examining six operations, previously considered under conventional wisdom as either special or conventional. It rejects much of the myth which surrounds these operations and offers a revisionist interpretation of the necessary and sufficient conditions for success in these endeavors. The thesis then goes on to compare cases of these special units using organizational theory to determine the sources of integration. The cases examined indicate a causal relationship between organizational factors related to command and training and a special units ability to integrate with GPF. The thesis concludes by recommending the realignment of select SOF in the current force structure to better meet the challenges of future integrated operations.

Special Operations Forces (SOF) are frequently employed to conduct missions not within the limited boundaries of unilateral special operations. These operations often involve cooperation with general purpose forces (GPF) and are often under their command. This thesis argues that these cooperative efforts are better examined as integrated operations rather than special or conventional operations. These operations require SOF to conduct specialized tasks facilitating the introduction of follow-on GPF to complete the mission. This thesis develops a theory of integrated operations by examining six operations, previously considered under conventional wisdom as either special or conventional. It rejects much of the myth which surrounds these operations and offers a revisionist interpretation of the necessary and sufficient conditions for success in these endeavors. The thesis then goes on to compare cases of these special units using organizational theory to determine the sources of integration. The cases examined indicate a causal relationship between organizational factors related to command and training and a special units ability to integrate with GPF. The thesis concludes by recommending the realignment of select SOF in the current force structure to better meet the challenges of future integrated operations.

We study the generalization of split and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference of two convex sets with specific geometric structures. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split and intersection cuts for several classes of sets. In particular, we give simple formulas for split cuts for essentially all convex sets described by a single quadratic inequality and for more general intersection cuts for a wide variety of convex quadratic sets.

This study develops a prototype model which can be used to allocate strategic sealift resources in crisis deployments. The first part of the model is a GAMS-Based Integer Program that extends a classic network flow optimization model developed by Dantzig and Fulkerson. The second part uses a Fortran program to convert the GAMS output into ship schedules. Using intelligent reduction methods, the formulation reduces the number of constraints by 60-70% and the number of variables and nonzero elements in the matrix by 90-99%. Results of this study indicate integer programming with these reduction methods is a viable alternative to modelling sealift as continuous flow variables.

This study develops a prototype model which can be used to allocate strategic sealift resources in crisis deployments. The first part of the model is a GAMS-Based Integer Program that extends a classic network flow optimization model developed by Dantzig and Fulkerson. The second part uses a Fortran program to convert the GAMS output into ship schedules. Using intelligent reduction methods, the formulation reduces the number of constraints by 60-70% and the number of variables and nonzero elements in the matrix by 90-99%. Results of this study indicate integer programming with these reduction methods is a viable alternative to modelling sealift as continuous flow variables.

Lecture from a Business Mathematics course (2005 - 6) by KIT

"NPS-55-82-009"--Cover

Thesis advisor(s): Dell, Robert F

Thesis advisor(s): Dell, Robert F

We accomplished our objectives, successfully implementing the Neighborhood Covering Heuristic (NCH) for solving the mixed integer programming problems. NCH is a unique, proprietary approach with several ground-breaking advantages. We completed a series of comparisons between NCH and the standard Branch and Bound (BAB) approach. Using the tunable parameters available within NCH, we created ten variants. We randomly generated sets of MIP problems, where the numbers of integer and binary variables, and constraints were varied in a systematic way. Then we applied both BAB and our ten variants of NCH to each of the generated problems. We obtained several significant results. First, when other parameters are fixed, the number of integer and binary variables in a random MIP problem does not have an exponential impact on the time required to find a feasible solution using NCH. Second, the performance data show that NCH produces feasible solutions significantly faster than BAB. Moreover the variance in time to produce a feasible solution is smaller in NCH. This reduction in variance and the resulting greater predictability of time to first solution will be extremely attractive to logistic companies, consultants, and useful directly in the Navy COMPASS program specifically and for Navy optimization needs in general.

We describe an approach for accomplishing the high-level mission planning required for a heterogeneous team of autonomous vehicles performing mine countermeasure (MCM) survey missions in multiple areas. The high-level mission scheduling and waterspace management requires sequencing the order and location of lower-level MCM tasks to be completed by each vehicle in the heterogeneous team: unmanned surface vessels (USVs) and unmanned underwater vehicles (UUVs). We propose solving this complex sequencing operation by leveraging unique information processing, communication, refueling, and planning windows that form constraints within the system within a formal scheduling optimization framework known as mixed-integer linear programming. We pose the problem using a mixed-integer linear programming optimization framework, compare several complexity reduction heuristics to the full optimization, and include methods to account for relative uncertainty in the duration of planned tasks in such a manner to balance the risk of schedule slips or conservative and sub-optimal schedule.

This paper presents a new implicit enumeration algorithm for solving the pure integer linear programming problem. The theory of equivalent integer programming problems is first used to reformulate the problem. A technique, originally used with particular success in the bound-and-scan algorithm to deal with only a subset of the variables, is extended to all of the variables in the restructured problem. In addition to the resulting basic enumeration scheme, the algorithm includes a scanning procedure and a method for identifying constraints which become redundant during the course of the algorithm. Computational experience on standard test problems is reported.

This paper presents a new implicit enumeration algorithm for solving the pure integer linear programming problem. The theory of equivalent integer programming problems is first used to reformulate the problem. A technique, originally used with particular success in the bound-and-scan algorithm to deal with only a subset of the variables, is extended to all of the variables in the restructured problem. In addition to the resulting basic enumeration scheme, the algorithm includes a scanning procedure and a method for identifying constraints which become redundant during the course of the algorithm. Computational experience on standard test problems is reported.

ADA743746

Motivated by Bland's linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we discuss three closely-related natural augmentation rules for linear and integer-linear optimization. In several nice situations, we show that polynomially-many augmentation steps suffice to reach an optimum. In particular, when using "discrete steepest-descent augmentations" (i.e., directions with the best ratio of cost improvement per unit 1-norm length), we show that the number of augmentation steps is bounded by the number of elements in the Graver basis of the problem matrix, giving the first ever strongly polynomial-time algorithm for $N$-fold integer-linear optimization. Our results also improve on what is known for such algorithms in the context of linear optimization (e.g., generalizing the bounds of Kitahara and Mizuno for the number of steps in the simplex method) and are closely related to research on the diameters of polytopes and the search for a strongly polynomial-time simplex or augmentation algorithm.

Motivated by Bland's linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we discuss three closely-related natural augmentation rules for linear and integer-linear optimization. In several nice situations, we show that polynomially-many augmentation steps suffice to reach an optimum. In particular, when using "discrete steepest-descent augmentations" (i.e., directions with the best ratio of cost improvement per unit 1-norm length), we show that the number of augmentation steps is bounded by the number of elements in the Graver basis of the problem matrix, giving the first ever strongly polynomial-time algorithm for $N$-fold integer-linear optimization. Our results also improve on what is known for such algorithms in the context of linear optimization (e.g., generalizing the bounds of Kitahara and Mizuno for the number of steps in the simplex method) and are closely related to research on the diameters of polytopes and the search for a strongly polynomial-time simplex or augmentation algorithm.

Motivated by Bland's linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we discuss three closely-related natural augmentation rules for linear and integer-linear optimization. In several nice situations, we show that polynomially-many augmentation steps suffice to reach an optimum. In particular, when using "discrete steepest-descent augmentations" (i.e., directions with the best ratio of cost improvement per unit 1-norm length), we show that the number of augmentation steps is bounded by the number of elements in the Graver basis of the problem matrix, giving the first ever strongly polynomial-time algorithm for $N$-fold integer-linear optimization. Our results also improve on what is known for such algorithms in the context of linear optimization (e.g., generalizing the bounds of Kitahara and Mizuno for the number of steps in the simplex method) and are closely related to research on the diameters of polytopes and the search for a strongly polynomial-time simplex or augmentation algorithm.

Motivated by Bland's linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we discuss three closely-related natural augmentation rules for linear and integer-linear optimization. In several nice situations, we show that polynomially-many augmentation steps suffice to reach an optimum. In particular, when using "discrete steepest-descent augmentations" (i.e., directions with the best ratio of cost improvement per unit 1-norm length), we show that the number of augmentation steps is bounded by the number of elements in the Graver basis of the problem matrix, giving the first ever strongly polynomial-time algorithm for $N$-fold integer-linear optimization. Our results also improve on what is known for such algorithms in the context of linear optimization (e.g., generalizing the bounds of Kitahara and Mizuno for the number of steps in the simplex method) and are closely related to research on the diameters of polytopes and the search for a strongly polynomial-time simplex or augmentation algorithm.

Slides used for group presentation.

Slides used for group presentation.

Slides used for group presentation.

Slides used for group presentation.

Slides used for group presentation.

Bibliography: p. 41-42

This paper presents Multi-Objective Integer Non-Linear Programming (MOINLP) involving Negative Total Float (NTF) for improving the basic model of Multi-Objective Programming (MOP) in case the optimization of the additional cost for Project Scheduling Compression (PSC). Using the basic MOP to solve the more complex problems is a challenging task. We suspect that Negative Total Float (NTF) having an indication to make the basic MOP to solve the more general case, both simple and complex of PSC. The purpose of this research is identifying the conflicting objectives in PSC problem using NTF and improving MOINLP by involving the NTF parameter to solve the PSC problem. The Solver Application, which is an add-in of MS Excel, is used to perform optimization process to the model developed. The results show that NTF has an important role to identify the conflicting objectives in PSC. We define NTF is an automatic maximum value of the activity duration reduction to achieve due date of PSC. Furthermore, the use of NTF as a constraint in MOINLP can solve the more general case for both simple and complex PSC problem. Base on the condition, we state that the basic MOP is still significant to solve the PSC complex problems using MOINLP as a sophisticated MOP technique. 

Thesis advisor, Richard E. Rosenthal

In this paper we present methods for exemplar based clustering with outlier selection based on the facility location formulation. Given a distance function and the number of outliers to be found, the methods automatically determine the number of clusters and outliers. We formulate the problem as an integer program to which we present relaxations that allow for solutions that scale to large data sets. The advantages of combining clustering and outlier selection include: (i) the resulting clusters tend to be compact and semantically coherent (ii) the clusters are more robust against data perturbations and (iii) the outliers are contextualised by the clusters and more interpretable, i.e. it is easier to distinguish between outliers which are the result of data errors from those that may be indicative of a new pattern emergent in the data. We present and contrast three relaxations to the integer program formulation: (i) a linear programming formulation (LP) (ii) an extension of affinity propagation to outlier detection (APOC) and (iii) a Lagrangian duality based formulation (LD). Evaluation on synthetic as well as real data shows the quality and scalability of these different methods.

Projection from a Business Mathematics course (2005 - 6) by KIT

Due to high system capacity requirement, 3GPP Long Term Evolution (LTE) is likely to adopt frequency reuse factor 1 at the cost of suffering severe inter-cell interference (ICI). One of combating ICI strategies is network cooperation of resource allocation (RA). For LTE uplink RA, requiring all the subcarriers to be allocated adjacently complicates the RA problem greatly. This paper investigates the joint multi-cell RA problem for LTE uplink. We model the uplink RA and ICI mitigation problem using pure binary-integer programming (BIP), with integrative consideration of all users' channel state information (CSI). The advantage of the pure BIP model is that it can be solved by branch-and-bound search (BBS) algorithm or other BIP solving algorithms, rather than resorting to exhaustive search. The system-level simulation results show that it yields 14.83% and 22.13% gains over single-cell optimal RA in average spectrum efficiency and 5th percentile of user throughput, respectively.

In this paper, some deficiencies of a method proposed by Burdet and Johnson in 1977 for solving integer programming problems are discussed. Examples where the algorithm fails to solve the IP and ways to fix these errors are given.

A nonlinear integer programming model for expanding the transportation system of an underdeveloped country is presented. The model uses integer 0-1 decision variables. The basic model has linear constraints and a nonlinear objective function. Some special situations and extensions to the model are presented. The benefits being maximized in the objective function are discussed, as are the problems of parameterization and suboptimization. A solution procedure for the model is suggested, but an efficient algorithm is not available for solving the model. Some areas for future research are also suggested.

An algorithm for the solution of integer linear programming problems is presented and programmed in Fortran IV for use off digital computers. The program incorporates an optional feature which provides all existing alternative optimal solutions. Solutions, computation times, and iteration requirements for each of thirteen test problems are summarized and discussed.