Abstract. We describe FATCOP, a new parallel mixed integer program solver written in PVM. The implementation uses the Condor resource management system to provide a virtual machine composed of otherwise idle computers. The solver differs from previous parallel branch-and-bound codes by implementing a general purpose parallel mixed integer programming algorithm in an opportunistic multiple processor environment, as opposed to a conventional dedicated environment. It shows how to make effective use of resources as they become available while ensuring the program tolerates resource retreat. The solver performs well on test problems arising from real applications and is particularly useful for solving long running hard mixed integer programming problems.

We investigate strong inequalities for mixed 0-1 integer programs derived from flow cover inequalities. Flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities. However, because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients, lifting of flow cover inequalities is computationally very demanding. We present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and computational results that justify the effectiveness of our lifting procedures. 1 Introduction A mixed integer program (MIP) with binary integer variables (BMIP) is the appropriate mathematical model for many practical optimization problems. This model is used, for example, for facility location problems, distribution problems, network design problems and more generally when fixed or concave costs are re...

We study the mixed--integer knapsack polyhedron, that is, the convex hull of the mixed--integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet--defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed--integer programming.

We investigate lifting, i.e., the process of taking a valid inequality for a polyhedron and extending it to a valid inequality in a higher dimensional space. Lifting is usually applied sequentially, that is, variables in a set are lifted one after the other. This may be computationally unattractive since it involves the solution of an optimization problem to compute a lifting coefficient for each variable. To relieve this computational burden, we study sequence independent lifting, which only involves the solution of one optimization problem. We show that if a certain lifting function is superadditive, then the lifting coefficients are independent of the lifting sequence. We introduce the idea of valid superadditive lifting functions to obtain good aproximations to maximum lifting. We apply these results to strengthen Balas' lifting theorem for cover inequalities and to produce lifted flow cover inequalities for a single node flow problem. December 1995 Revised July 1997 Revised Janua...

The use of floating-point calculations limits the accuracy of solutions obtained by standard LP software. We present a simplex-based algorithm that returns exact rational solutions, taking advantage of the speed of floating-point calculations and attempting to minimize the operations performed in rational arithmetic. Extensive computational results are presented.

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In the Median Cycle Problem the aim is to determine a simple cycle containing a subset of vertices of a graph, while considering two types of cost: routing costs associated with the cycle itself, and the cost of assigning vertices not on the cycle to vertices of the cycle. Two versions of the problem are investigated. In the rst, the objective function is the sum of routing and assignment costs. In the second, routing costs are minimized, subject to an upper bound on the total assignment cost. The two versions are formulated as integer linear programs. The polyhedral structure of the rst model is analyzed and the second model is strengthened through the introduction of additional valid inequalities. Separation procedures are developed. Heuristic procedures and an exact branch-and-cut algorithm are described. Computational results on benchmark instances and on randomly generated problems con rm the e ciency of the proposed algorithms.

Dual fractional cutting plane algorithms, in which cutting planes are used to iteratively tighten a linear relaxation of an integer program, are wellknown and form the basis of the highly successful branch-and-cut method. It is rather less well-known that various primal cutting plane algorithms were developed in the 1960s, for example by Young. In a primal algorithm, the main role of the cutting planes is to enable a feasible solution to the original problem to be improved. Research on these algorithms has been almost nonexistent. In this paper we argue for a re-examination of these primal methods. We describe a new primal algorithm for pure 0-1 problems based on strong valid inequalities and give some encouraging computational results. Possible extensions to the case of general mixed-integer programs are also discussed.

Abstract. Recent developments in integer–programming software systems have tremendously improved our ability to solve large–scale instances. We review the major algorithmic components of state–of–the–art solvers and discuss the options available to users to adjust the behavior of these solvers when default settings do not achieve the desired performance level. Furthermore, we highlight advances towards integrated modeling and solution environments. We conclude with a discussion of model characteristics and substructures that pose challenges for integer–programming software systems and a perspective on features we may expect to see in these systems in the near future. 1.

In a recent paper, the authors argued for a re-examination of primal cutting plane algorithms, in which cutting planes are used to enable a feasible solution to the original problem to be improved. This led to the idea of primal separation algorithms, which are similar to standard separation algorithms but tailored to the primal context. In this paper we examine the complexity of primal separation for several well-known classes of inequalities for various important combinatorial optimization problems. It turns out that, in several important cases, the primal separation problem can be solved more easily than the standard one. This is a strong argument in favour of the primal approach. One of the most striking results is that there is an exact polynomialtime primal separation algorithm for a generalization of the so-called Chvatal comb inequalities for the TSP. No polynomial-time algorithm is known for the standard separation problem. Key Words: integer programming, cutting planes, separation, primal algorithms, Travelling Salesman Problem. 1