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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 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 consider the problem of finding an n-agent joint-policy for the optimal finite-horizon control of a decentralized Pomdp (Dec-Pomdp). This is a problem of very high complexity (NEXP-hard in n >= 2). In this paper, we propose a new mathematical programming approach for the problem. Our approach is based on two ideas: First, we represent each agent's policy in the sequence-form and not in the tree-form, thereby obtaining a very compact representation of the set of joint-policies. Second, using this compact representation, we solve this problem as an instance of combinatorial optimization for which we formulate a mixed integer linear program (MILP). The optimal solution of the MILP directly yields an optimal joint-policy for the Dec-Pomdp. Computational experience shows that formulating and solving the MILP requires significantly less time to solve benchmark Dec-Pomdp problems than existing algorithms. For example, the multi-agent tiger problem for horizon 4 is solved in 72 secs with the MILP whereas existing algorithms require several hours to solve it.

We propose a novel approach to automatically tracking cell populations in time-lapse images. To account for cell occlusions and overlaps, we introduce a robust method that generates an over-complete set of competing detection hypotheses. We then perform detection and tracking simultaneously on these hypotheses by solving to optimality an integer program with only one type of flow variables. This eliminates the need for heuristics to handle missed detections due to occlusions and complex morphology. We demonstrate the effectiveness of our approach on a range of challenging sequences consisting of clumped cells and show that it outperforms state-of-the-art techniques.

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 article is from BMC Proceedings , volume 7 . Abstract A major challenge in microarray classification is that the number of features is typically orders of magnitude larger than the number of examples. In this paper, we propose a novel feature filter algorithm to select the feature subset with maximal discriminative power and minimal redundancy by solving a quadratic objective function with binary integer constraints. To improve the computational efficiency, the binary integer constraints are relaxed and a low-rank approximation to the quadratic term is applied. The proposed feature selection algorithm was extended to solve multi-task microarray classification problems. We compared the single-task version of the proposed feature selection algorithm with 9 existing feature selection methods on 4 benchmark microarray data sets. The empirical results show that the proposed method achieved the most accurate predictions overall. We also evaluated the multi-task version of the proposed algorithm on 8 multi-task microarray datasets. The multi-task feature selection algorithm resulted in significantly higher accuracy than when using the single-task feature selection methods.

This article is from PLoS ONE , volume 9 . Abstract In this paper, we consider the Minimum Reaction Insertion (MRI) problem for finding the minimum number of additional reactions from a reference metabolic network to a host metabolic network so that a target compound becomes producible in the revised host metabolic network in a Boolean model. Although a similar problem for larger networks is solvable in a flux balance analysis (FBA)-based model, the solution of the FBA-based model tends to include more reactions than that of the Boolean model. However, solving MRI using the Boolean model is computationally more expensive than using the FBA-based model since the Boolean model needs more integer variables. Therefore, in this study, to solve MRI for larger networks in the Boolean model, we have developed an efficient Integer Programming formalization method in which the number of integer variables is reduced by the notion of feedback vertex set and minimal valid assignment. As a result of computer experiments conducted using the data of metabolic networks of E. coli and reference networks downloaded from the Kyoto Encyclopedia of Genes and Genomes (KEGG) database, we have found that the developed method can appropriately solve MRI in the Boolean model and is applicable to large scale-networks for which an exhaustive search does not work. We have also compared the developed method with the existing connectivity-based methods and FBA-based methods, and show the difference between the solutions of our method and the existing methods. A theoretical analysis of MRI is also conducted, and the NP-completeness of MRI is proved in the Boolean model. Our developed software is available at “http://sunflower.kuicr.kyoto-u.ac.jp/~rogi/minRect/minRect.html.”

We develop Integer Programming (IP) solutions for some special college admission problems arising from the Hungarian higher education admission scheme. We focus on four special features, namely the solution concept of stable score-limits, the presence of lower and common quotas, and paired applications. We note that each of the latter three special feature makes the college admissions problem NP-hard to solve. Currently, a heuristic based on the Gale-Shapley algorithm is being used in the application. The IP methods that we propose are not only interesting theoretically, but may also serve as an alternative solution concept for this practical application, and also for other ones.

The development is documented of a heuristic method for the solution of pure integer linear programming problems. The procedure draws its methodology from the ideas of Hooke and Jeeves type 1 and 2 exploratory searches, greedy procedures, and neighborhood searches. It uses an efficient rounding method to obtain its first feasible integer point from the optimal continuous solution obtained via the simplex method. Since this method is based entirely on simple addition or subtraction of one to each variable of a point in n-space and the subsequent comparison of candidate solutions to a given set of constraints, it facilitates significant complexity improvements over existing techniques. It also obtains the same optimal solution found by the branch-and-bound technique in 44 of 45 small to moderate size test problems. Two example problems are worked in detail to show the inner workings of the method. Furthermore, using an established weighted scheme for comparing computational effort involved in an algorithm, a comparison of this algorithm is made to the more established and rigorous branch-and-bound method. A computer implementation of the procedure, in PC compatible Pascal, is also presented and discussed.

The development is documented of a heuristic method for the solution of pure integer linear programming problems. The procedure draws its methodology from the ideas of Hooke and Jeeves type 1 and 2 exploratory searches, greedy procedures, and neighborhood searches. It uses an efficient rounding method to obtain its first feasible integer point from the optimal continuous solution obtained via the simplex method. Since this method is based entirely on simple addition or subtraction of one to each variable of a point in n-space and the subsequent comparison of candidate solutions to a given set of constraints, it facilitates significant complexity improvements over existing techniques. It also obtains the same optimal solution found by the branch-and-bound technique in 44 of 45 small to moderate size test problems. Two example problems are worked in detail to show the inner workings of the method. Furthermore, using an established weighted scheme for comparing computational effort involved in an algorithm, a comparison of this algorithm is made to the more established and rigorous branch-and-bound method. A computer implementation of the procedure, in PC compatible Pascal, is also presented and discussed.

The development is documented of a heuristic method for the solution of pure integer linear programming problems. The procedure draws its methodology from the ideas of Hooke and Jeeves type 1 and 2 exploratory searches, greedy procedures, and neighborhood searches. It uses an efficient rounding method to obtain its first feasible integer point from the optimal continuous solution obtained via the simplex method. Since this method is based entirely on simple addition or subtraction of one to each variable of a point in n-space and the subsequent comparison of candidate solutions to a given set of constraints, it facilitates significant complexity improvements over existing techniques. It also obtains the same optimal solution found by the branch-and-bound technique in 44 of 45 small to moderate size test problems. Two example problems are worked in detail to show the inner workings of the method. Furthermore, using an established weighted scheme for comparing computational effort involved in an algorithm, a comparison of this algorithm is made to the more established and rigorous branch-and-bound method. A computer implementation of the procedure, in PC compatible Pascal, is also presented and discussed.

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

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

Journal of Research of the National Bureau of Standards

In recent years, the embedding approach for switched optimal control problems has been developed in a series of papers. However, the embedding approach, which advantageously converts the hybrid optimal control problem to a classical nonlinear optimization, has not been extensively compared to alternative approaches. The goal of this paper is thus to compare the embedding approach to multi-parametric programming, mixed-integer programming, gradient-descent based methods, and CPLEX in the context of five recently published examples. A sixth example, an autonomous switched 11-region linear system, is used to compare a hybrid minimum principle method and traditional numerical programming. For a given performance index for each case, cost and solution times are presented. It is shown that there are numerical advantages of the embedding approach: lower performance index cost (except in some instances when autonomous switches are present), generally faster solution time, and convergence to a solution when other methods may fail. In addition, the embedding method requires no ad hoc assumptions (e.g., predetermined mode sequences) or specialized control models. Theoretical advantages of the embedding approach over the other methods are also described: guaranteed existence of a solution under mild conditions, convexity of the embedded hybrid optimization problem (under the customary conditions on the performance index), solvability with traditional techniques (e.g., sequential quadratic programming) avoiding the combinatorial complexity in the number of modes/discrete variables of mixed-integer programming, applicability to affine nonlinear systems, and no need to explicitly assign discrete/mode variables to autonomous switches.

Inspired by a paper of R. W. Rosenthal, we investigate generalized Nash-equilibria of integer programming games. We show that generalized Nash-equilibria always exist and are related to an optimal solution of a so-called N-fold integer program. This link allows us to establish some polynomial time complexity results about solving this optimization problem and its inverse counter-part.

Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter. In 2014, Mnich and Wiese initiated a systematic study in this direction. In this paper we continue this study and show that several additional cases of fundamental scheduling problems are fixed parameter tractable for some natural parameters. Our main tool is n-fold integer programming, a recent variable dimension technique which we believe to be highly relevant for the parameterized complexity community. This paper serves to showcase and highlight this technique. Specifically, we show the following four scheduling problems to be fixed-parameter tractable, where p max is the maximum processing time of a job and w max is the maximum weight of a job: - Makespan minimization on uniformly related machines $(Q||C_{max} )$ parameterized by $p_{max}$, - Makespan minimization on unrelated machines $(R||C_{max} )$ parameterized by $p_{max}$ and the number of kinds of machines, - Sum of weighted completion times minimization on unrelated machines $(R|| \sum w_i C_i )$ parameterized by $p_{max} + w_{max}$ and the number of kinds of machines, - The same problem, $(R|| \sum w_i C_i),$ parameterized by the number of distinct job times and the number of machines.

This paper addresses the single-item single-stocking location stochastic lot sizing problem under the $(s, S) $ policy. We first present a mixed integer non-linear programming (MINLP) formulation for determining near-optimal $(s, S)$ policy parameters. To tackle larger instances, we then combine the previously introduced MINLP model and a binary search approach. These models can be reformulated as mixed integer linear programming (MILP) models which can be easily implemented and solved by using off-the-shelf optimisation software. Computational experiments demonstrate that optimality gaps of these models are around $0.3\%$ of the optimal policy cost and computational times are reasonable.

This paper addresses the single-item single-stocking location stochastic lot sizing problem under the $(s, S) $ policy. We first present a mixed integer non-linear programming (MINLP) formulation for determining near-optimal $(s, S)$ policy parameters. To tackle larger instances, we then combine the previously introduced MINLP model and a binary search approach. These models can be reformulated as mixed integer linear programming (MILP) models which can be easily implemented and solved by using off-the-shelf optimisation software. Computational experiments demonstrate that optimality gaps of these models are around $0.3\%$ of the optimal policy cost and computational times are reasonable.

Bibliography: pt. 1, p. 46-47; pt. 2, p. 48

A quadratic program (QP) is a well-studied fundamental NP-hard optimization problem which optimizes a quadratic objective over a set of linear constraints. In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This is done via the reformulation of QP as a linear complementary problem, and the use of binary variables together with some fundamental results on the solution of perturbed linear systems, to model the complementary constraints. Reformulating non-convex QPs as MILPs provides an advantageous way to obtain global solutions as it allows to use current state-of-the-art MILP solvers. To illustrate, we compare the performance of our solution approach with the current benchmark global QP solver quadprogBB on a large variety of QP test instances. The MATLAB code, called quadprogIP, and the instances used to perform these numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.

Multi-agent differential games are important and useful tools for analyzing many practical problems. With the recent surge of interest in using UAVs for civil purposes, the importance and urgency of developing tractable multi-agent analysis techniques that provide safety and performance guarantees is at an all-time high. Hamilton-Jacobi (HJ) reachability has successfully provided safety guarantees to small-scale systems and is flexible in terms of system dynamics. However, the exponential complexity scaling of HJ reachability prevents its direct application to large scale problems when the number of vehicles is greater than two. In this paper, we overcome the scalability limitations of HJ reachability by using a mixed integer program that exploits the properties of HJ solutions to provide higher-level control logic. Our proposed method provides safety guarantee for three-vehicle systems -- a previously intractable task for HJ reachability -- without incurring significant additional computation cost. Furthermore, our method is scalable beyond three vehicles and performs significantly better by several metrics than an extension of pairwise collision avoidance to multi-vehicle collision avoidance. We demonstrate our proposed method in simulations.

Extraction of events and understanding related temporal expression among them is a major challenge in natural language processing. In longer texts, processing on sentence-by-sentence or expression-by-expression basis often fails, in part due to the disregard for the consistency of the processed data. We present an ensemble method, which reconciles the output of multiple classifiers for temporal expressions, subject to consistency constraints across the whole text. The use of integer programming to enforce the consistency constraints globally improves upon the best published results from the TempEval-3 Challenge considerably.

Extraction of events and understanding related temporal expression among them is a major challenge in natural language processing. In longer texts, processing on sentence-by-sentence or expression-by-expression basis often fails, in part due to the disregard for the consistency of the processed data. We present an ensemble method, which reconciles the output of multiple classifiers for temporal expressions, subject to consistency constraints across the whole text. The use of integer programming to enforce the consistency constraints globally improves upon the best published results from the TempEval-3 Challenge considerably.

We study Graver test sets for families of linear multi-stage stochastic integer programs with varying number of scenarios. We show that these test sets can be decomposed into finitely many ``building blocks'', independent of the number of scenarios, and we give an effective procedure to compute these building blocks. The paper includes an introduction to Nash-Williams' theory of better-quasi-orderings, which is used to show termination of our algorithm. We also apply this theory to finiteness results for Hilbert functions.

Bibliography: p. 41-42

Established through the Marine Security Guard (MSG) Program during the 1940s, the Marine Corps and Department of State have shared a partnership of providing critical security to designated diplomatic facilities worldwide. Approximately 250 Marines execute permanent charge-of-station orders within the program five times every year to support personnel manning requirements. Are these Marines being sent to the right location? Is one embassy unintentionally staffed with a disproportionate quality of MGSs? Is there a better metric to measure and assign Marines based on a decision-maker_s preference? The current assignment process is manpower-intensive and involves more than 15 personnel across three levels of command. At present, there is no formal methodology to quantity or measure how well MSGs are being assigned. The purpose of this research is to provide Marine Corps Embassy Security Group (MCESG) Headquarters senior leaders with an alternative method to complement the current assignment process by equitably distributing the quality of MSGs using integer programming. The results of this research support an improvement by up to 96% of distributing quality using the sum of squared differences across each region. The impact of using these alternative methods can be expected to significantly decrease MCESG assignment man-hours.

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. 

In the minimum common string partition (MCSP) problem two related input strings are given. "Related" refers to the property that both strings consist of the same set of letters appearing the same number of times in each of the two strings. The MCSP seeks a minimum cardinality partitioning of one string into non-overlapping substrings that is also a valid partitioning for the second string. This problem has applications in bioinformatics e.g. in analyzing related DNA or protein sequences. For strings with lengths less than about 1000 letters, a previously published integer linear programming (ILP) formulation yields, when solved with a state-of-the-art solver such as CPLEX, satisfactory results. In this work, we propose a new, alternative ILP model that is compared to the former one. While a polyhedral study shows the linear programming relaxations of the two models to be equally strong, a comprehensive experimental comparison using real-world as well as artificially created benchmark instances indicates substantial computational advantages of the new formulation.

In this paper we propose a fast optimization algorithm for approximately minimizing convex quadratic functions over the intersection of affine and separable constraints (i.e., the Cartesian product of possibly nonconvex real sets). This problem class contains many NP-hard problems such as mixed-integer quadratic programming. Our heuristic is based on a variation of the alternating direction method of multipliers (ADMM), an algorithm for solving convex optimization problems. We discuss the favorable computational aspects of our algorithm, which allow it to run quickly even on very modest computational platforms such as embedded processors. We give several examples for which an approximate solution should be found very quickly, such as management of a hybrid-electric vehicle drivetrain and control of switched-mode power converters. Our numerical experiments suggest that our method is very effective in finding a feasible point with small objective value; indeed, we find that in many cases, it finds the global solution.

In this paper we propose a fast optimization algorithm for approximately minimizing convex quadratic functions over the intersection of affine and separable constraints (i.e., the Cartesian product of possibly nonconvex real sets). This problem class contains many NP-hard problems such as mixed-integer quadratic programming. Our heuristic is based on a variation of the alternating direction method of multipliers (ADMM), an algorithm for solving convex optimization problems. We discuss the favorable computational aspects of our algorithm, which allow it to run quickly even on very modest computational platforms such as embedded processors. We give several examples for which an approximate solution should be found very quickly, such as management of a hybrid-electric vehicle drivetrain and control of switched-mode power converters. Our numerical experiments suggest that our method is very effective in finding a feasible point with small objective value; indeed, we find that in many cases, it finds the global solution.

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

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

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

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.

In this article we study a broad class of integer programming problems in variable dimension. We show that these so-termed {\em n-fold integer programming problems} are polynomial time solvable. Our proof involves two heavy ingredients discovered recently: the equivalence of linear optimization and so-called directed augmentation, and the stabilization of certain Graver bases. We discuss several applications of our algorithm to multiway transportation problems and to packing problems. One important consequence of our results is a polynomial time algorithm for the $d$-dimensional integer transportation problem for long multiway tables. Another interesting application is a new algorithm for the classical cutting stock problem.

In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver complexity of the incidence matrix of the complete bipartite graph $K_{3,m}$ satisfies $g(m)=\Omega(2^m)$, with $g(m)\geq 17\cdot 2^{m-3}-7$ for every $m>3$ .

Optimal distribution of power among generating units to meet a specific demand subject to system constraints is an ongoing research topic in the power system community. The problem, even in a static setting, turns out to be hard to solve with conventional optimization methods owing to the consideration of valve-point effects which make the cost function nonsmooth and nonconvex. This difficulty gave rise to the proliferation of population-based global heuristics in order to address the multi-extremal and nonsmooth problem. In this paper, we address the economic load dispatch problem (ELDP) with valve-point effect in its classic formulation where the cost function for each generator is expressed as the sum of a quadratic term and a rectified sine term. We propose two methods that resort to piecewise-quadratic surrogate cost functions, yielding surrogate problems that can be handled by mixed-integer quadratic programming (MIQP) solvers. The first method shows that the global solution of the ELDP can often be found by using a fixed and very limited number of quadratic pieces in the surrogate cost function. The second method adaptively builds piecewise-quadratic surrogate under-estimations of the ELDP cost function, yielding a sequence of surrogate MIQP problems. It is shown that any limit point of the sequence of MIQP solutions is a global solution of the ELDP. Moreover, numerical experiments indicate that the proposed methods outclass the state-of-the-art algorithms in terms of minimization value and computation time on practical instances.

The selection of branching variables is a key component of branch-and-bound algorithms for solving Mixed-Integer Programming (MIP) problems since the quality of the selection procedure is likely to have a significant effect on the size of the enumeration tree. State-of-the-art procedures base the selection of variables on their "LP gains", which is the dual bound improvement obtained after branching on a variable. There are various ways of selecting variables depending on their LP gains. However, all methods are evaluated empirically. In this paper we present a theoretical model for the selection of branching variables. It is based upon an abstraction of MIPs to a simpler setting in which it is possible to analytically evaluate the dual bound improvement of choosing a given variable. We then discuss how the analytical results can be used to choose branching variables for MIPs, and we give experimental results that demonstrate the effectiveness of the method on MIPLIB 2010 "tree" instances where we achieve a 5% geometric average time and node improvement over the default rule of SCIP, a state-of-the-art MIP solver.

The selection of branching variables is a key component of branch-and-bound algorithms for solving Mixed-Integer Programming (MIP) problems since the quality of the selection procedure is likely to have a significant effect on the size of the enumeration tree. State-of-the-art procedures base the selection of variables on their "LP gains", which is the dual bound improvement obtained after branching on a variable. There are various ways of selecting variables depending on their LP gains. However, all methods are evaluated empirically. In this paper we present a theoretical model for the selection of branching variables. It is based upon an abstraction of MIPs to a simpler setting in which it is possible to analytically evaluate the dual bound improvement of choosing a given variable. We then discuss how the analytical results can be used to choose branching variables for MIPs, and we give experimental results that demonstrate the effectiveness of the method on MIPLIB 2010 "tree" instances where we achieve a 5% geometric average time and node improvement over the default rule of SCIP, a state-of-the-art MIP solver.

The selection of branching variables is a key component of branch-and-bound algorithms for solving Mixed-Integer Programming (MIP) problems since the quality of the selection procedure is likely to have a significant effect on the size of the enumeration tree. State-of-the-art procedures base the selection of variables on their "LP gains", which is the dual bound improvement obtained after branching on a variable. There are various ways of selecting variables depending on their LP gains. However, all methods are evaluated empirically. In this paper we present a theoretical model for the selection of branching variables. It is based upon an abstraction of MIPs to a simpler setting in which it is possible to analytically evaluate the dual bound improvement of choosing a given variable. We then discuss how the analytical results can be used to choose branching variables for MIPs, and we give experimental results that demonstrate the effectiveness of the method on MIPLIB 2010 "tree" instances where we achieve a 5% geometric average time and node improvement over the default rule of SCIP, a state-of-the-art MIP solver.

Bibliography: leaves 20-21

Bibliography: l. 19-20

Bibliography: l. 19-20

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

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