An integer linear programming approach for bilinear integer programming

We introduce a new Integer Linear Programming (ILP) approach for solving Integer Programming (IP) problems with bilinear objectives and linear constraints. The approach relies on a series of ILP approximations of the bilinear IP. We compare this approach with standard linearization techniques on random instances and a set of real-world product bundling problems.     

Experiments in integer programming

A hybrid algorithm to solve large scale zero–one integer programming problems has been developed. The algorithm combines branch-and-bound, enumeration and cutting plane techniques. Mixed-integer cuts are generated in the initial phase of the algorithm and added to the L.P. Benders cuts are derived and used implicitly but, except for the cut from the initial LP, are not stored. The algorithm has been implemented on an experimental basis in MPSX/370 using its Extended Control Language and Algorithmic Tools. A computational study based on five well-known difficult test problems and on three practical problems with up to 2000 zer–one variables shows that the hybrid code compares favorably with MIP/370 and with results published for other algorithms.     

Robust integer programming

We provide a complexity classification of four variants of robust integer programming when the underlying Graver basis is given. We discuss applications to robust multicommodity flows and multiway statistical table problems, and describe an effective parametrization of robust integer programming.     

Generalized dynamic programming methods in integer programming

When regarded as a shortest route problem, an integer program can be seen to have a particularly simple structure. This allows the development of an algorithm for finding thek ^th best solution to an integer programming problem with max{O(kmn), O(k logk)} operations. Apart from its value in the parametric study of an optimal solution, the approach leads to a general integer programming algorithm consisting of (1) problem relaxation, (2) solution of the relaxed problem parametrically by dynamic programming, and (3) generation ofk ^th best solutions until a feasible solution is found. Elementary methods based on duality for reducingk for a given problem relaxation are then outlined, and some examples and computational aspects are discussed.     

Enumerative inequalities in integer programming

For a linear integer programming problem, thelocal information contained at an optimal solution of the continuous linear programming extension stems from the theory of L.P. solutions. This paper proposes the use ofenvironmental information (of a global nature but pertaining to the discrete vicinity of ), in order to isolate the set of integer solutions which may be considered as true candidates for the optimum. The concept ofenumerative inequalities is introduced and it is shown how it can be obtained in the context of the convex outer-domain theory of Balas, Young, et al.Generally speaking, enumerative inequalities can be made arbitrarily strong (deep), but at the cost of an increasing amount of work (i.e. enumeration) for their construction. In particular cases, however, very little global information can produce enumerative inequalities stronger than anyvalid cut.This paper was presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.     

Subadditive approaches in integer programming

Linear programming duality is well understood and the reduced cost of a column is frequently used in various algorithms. On the other hand, for integer programs it is not clear how to define a dual function even though the subadditive dual theory has been developed a long time ago. In this work we propose a family of computationally tractable subadditive dual functions for integer programs. We develop a solution methodology that computes an optimal primal solution and an optimal subadditive dual function. We present computational experiments, which show that the new algorithm is tractable.     

Modelling either-or relations in integer programming

Logical relations occur frequently in integer programming problems and are modelled by introducing binary variables in association with linear expressions. Applications requiring constraints involving precedence, exclusion, implication and other conditions give rise to the logical relations OR and IMPLIES in the models. These relations will be considered in this paper from a modelling point of view and formulations investigated for situations where the logical variables link sets of integer variables. Valid inequalities (cuts) that can be added to a model will be developed for a number of the formulations and the computational benefits of these cuts will be considered from an experimental point of view by considering the performance of sets of problem instances. New formulations and combinations of older established formulations will be considered. It will be contended that tight formulations may not always be the most successful.     

Developments in linear and integer programming

In this review we describe recent developments in linear and integer (linear) programming. For over 50 years Operational Research practitioners have made use of linear optimisation models to aid decision making and over this period the size of problems that can be solved has increased dramatically, the time required to solve problems has decreased substantially and the flexibility of modelling and solving systems has increased steadily. Large models are no longer confined to large computers, and the flexibility of optimisation systems embedded in other decision support tools has made on-line decision making using linear programming a reality (and using integer programming a possibility). The review focuses on recent developments in algorithms, software and applications and investigates some connections between linear optimisation and other technologies.     

Conflict analysis in mixed integer programming

Conflict analysis for infeasible subproblems is one of the key ingredients in modern SAT solvers. In contrast, it is common practice for today’s mixed integer programming solvers to discard infeasible subproblems and the information they reveal. In this paper, we try to remedy this situation by generalizing SAT infeasibility analysis to mixed integer programming.We present heuristics for branch-and-cut solvers to generate valid inequalities from the current infeasible subproblem and the associated branching information. SAT techniques can then be used to strengthen the resulting constraints. Extensive computational experiments show the potential of our method. Conflict analysis greatly improves the performance on particular models, while it does not interfere with the solving process on the other instances. In total, the number of required branching nodes on general MIP instances was reduced by 18% in the geometric mean, and the solving time was reduced by 11%. On infeasible MIPs arising in the context of chip verification and on a model for a particular combinatorial game, the number of nodes was reduced by 80%, thereby reducing the solving time by 50%.     

On convergence in mixed integer programming

Let ${P \subseteq {\mathbb R}^{m+n}}$ be a rational polyhedron, and let P _I be the convex hull of ${P \cap ({\mathbb Z}^m \times {\mathbb R}^n)}$ . We define the integral lattice-free closure of P as the set obtained from P by adding all inequalities obtained from disjunctions associated with integral lattice-free polyhedra in ${{\mathbb R}^m}$ . We show that the integral lattice-free closure of P is again a polyhedron, and that repeatedly taking the integral lattice-free closure of P gives P _I after a finite number of iterations. Such results can be seen as a mixed integer analogue of theorems by Chvátal and Schrijver for the pure integer case. One ingredient of our proof is an extension of a result by Owen and Mehrotra. In fact, we prove that for each rational polyhedron P, the split closures of P yield in the limit the set P _I .     

Sensitivity theorems in integer linear programming

We consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right-hand-side vector. Among our results, we show that, for any optimal solution to a linear program max{wx: Axb}, the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixA. Using this, we strengthen several integer programming proximity results of Blair and Jeroslow; Graver; and Wolsey. We also show that the Chvátal rank of a polyhedron {x: Axb} can be bounded above by a function of the matrixA, independent of the vectorb, a result which, as Blair observed, is equivalent to Blair and Jeroslow's theorem that each integer programming value function is a Gomory function.Supported by a grant from the Alexander von Humboldt Stiftung.Since September 1985: Department of Operations Research, Upson Hall, Cornell University, Ithaca, NY 14853, USA.Partially supported by the Sonderforschungbereich 21 (DFG), Institut für Ökonometrie und Operations Research of the University of Bonn, FR Germany.     

Aggregation of equations in integer programming

Several methods for replacing a system of equations (usually linear) in non-negative integer variables by a single equation have recently been proposed. In this paper an attempt is made to extend these results and to clarify the basic structures (in particular in the case of unbounded variables) with emphasis on existence rather than on the difficult problem of optimality. The direct aggregation into fewer constraints (not necessarily one) is studied (avoiding thus the problematic step by step approach) using integral matrices with relatively prime subdeterminants of the largest size. Most of the results are based on the fact that the intersection of a certain linear space with a certain set is bounded. This easy approach (which is characteristic to all known methods except one by Anthonisse and Bradley) may fail for unbounded variables or lead to large coefficients; therefore a method is proposed which under special circumstances may reduce the problem to the solvability of certain congruences.     

Algorithmic Chernoff-Hoeffding inequalities in integer programming

Proofs of classical Chernoff-Hoeffding bounds has been used to obtain polynomial-time implementations of Spencer's derandomization method of conditional probabilities on usual finite machine models: given m events whose complements are large deviations corresponding to weighted sums of n mutually independent Bernoulli trials. Raghavan's lattice approximation algorithm constructs for 0-1 weights, and integer deviation terms in O(mn)-time a point for which all events hold. For rational weighted sums of Bernoulli trials the lattice approximation algorithm or Spencer's hyperbolic cosine algorithm are deterministic precedures, but a polynomial-time implementation was not known. We resolve this problem with an O(mn²log $ {{mn}\over{\epsilon}} $)-time algorithm, whenever the probability that all events hold is at least ϵ > 0. Since such algorithms simulate the proof of the underlying large deviation inequality in a constructive way, we call it the algorithmic version of the inequality. Applications to general packing integer programs and resource constrained scheduling result in tight and polynomial-time approximations algorithms. © 1996 John Wiley & Sons, Inc.     

Parametric methods in integer linear programming

In contrast to methods of parametric linear programming which were developed soon after the invention of the simplex algorithm and are easily included as an extension of that method, techniques for parametric analysis on integer programs are not well known and require considerable effort to append them to an integer programming solution algorithm.The paper reviews some of the theory employed in parametric integer programming, then discusses algorithmic work in this area over the last 15 years when integer programs are solved by different methods. A summary of applications is included and the article concludes that parametric integer programming is a valuable tool of analysis awaiting further popularization.     

Cut search methods in integer programming

Cut search is a new approach for solving integer programs based on extending edges of a cone to probe the solution space for sets of hyperplanes that are proxies for solution points in the space. Once all proxy hyperplanes associated with a given point have been intersected by at least one of the extended edges, this point is included in a set of points to be examined for feasibility (algorithmically or by inspection). Thereupon, all edges of the cone are extended an additional distance to create a cut by passing a hyperplane through the endpoints of these extended edges.The flexibility of the cut search procedure permits a variety of strategies for exploring and cutting into the solution space. One useful version arises by taking the proxy hyperplanes to be members of a positive or semipositive coordinate system. Relative to such a system the procedure can be organized to reduce the set of vectors to be examined for feasibility and also to generate deeper cuts at the end of the edge probe.